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I. V. Savelyev 

PHYSICS 

A General Course 

ELECTRICITY 
AND MAGNETISM 

WAVES 

OPTICS 


Mir Publishers 
Moscow 




H. B. CABEJILEB 
KYPC OBIIJEH CDH3HKH 
TOM II 

ajIEKTPHHECTBO H 
MArHETH3M 

BOJIHBI 

OIITHKA 


H3flATEJlbCTBO «HAyKA* 
MOCKBA 



I. V. SAVELYEV 


PHYSICS 

A GENERAL COURSE 

(In three volumes) 


VOLUME II 

ELECTRICITY 
AND MAGNETISM 
WAVES 
OPTICS 



MIR Publishers 
MOSCOW 



Translated from Russian by G. Leib 


First published 1980 

Revised from the 1978 Russian edition 

Second printing 1985 

Third printing 1989 



Ha amnuucKOM H3biKe 


Printed in the Union of Soviet Socialist Republics 


ISBN 5-03 000902-7 
ISBN 5-03-000900-0 


© H3naTejibCTBo «Hayica», 1978 
© English translation, Mir Publishers, 1980 



PREFACE 


The main content of the present volume is the science of electro- 
magnetism and the science of waves (elastic, electromagnetic, and 
light). 

The International System of Units (SI) has been used throughout 
the book, although the reader is simultaneously acquainted with 
the Gaussian system. In addition to a list of symbols, the appendices 
at the end of the book give the units of electrical and magnetic 
quantities in the SI and in the Gaussian system of units, and also 
compare the form of the basic formulas of electromagnetism in both 
systems. 

The course is the result of twenty five year’s work in the Depart- 
ment of General Physics of the Moscow Institute of Engineering 
Physics. I am grateful to my colleagues and friends for their helpful 
discussions, criticism and advice in the course of the preparation 
of the book. 

The present course is intended above all for higher technical 
schools with an extended syllabus in physics. The material has 
been arranged, however, so that the book can be used as a teaching 
aid for higher technical schools with an ordinary syllabus simply 
by omitting some sections. 


Moscow, November, 1979 


Igor Savelyev 



CONTENTS 


PART I. 

CHAPTER 


-v 


CHAPTER 


CHAPTER 


CHAPTER 


Preface 5 


ELECTRICITY AND MAGNETISM 11 

1. ELECTRIC FIELD IN A VACUUM 

1.1. Electric Charge 11 

1.2. Coulomb’s Law 12 

1.3. Systems of Units 14 

1.4. Rationalized Form of Writing Formulas 16 

1.5. Electric Field. Field Strength 16 

1.6. Potential 20 

1.7. Interaction Energy of a System of Charges 24 

1.8. Relation Between Electric Field Strength and Potential 25 

1.9. Dipole 28 

1.10. Field of a System of Charges at Great Distances 33 

1.11. A Description of the Properties of Vector Fields 36 

1.12. Circulation and Curl of an Electrostatic Field 51 

1.13. Gauss’s Theorem 53 

1.14. Calculating Fields with the Aid of Gauss’s Theorem 55 

2. ELECTRIC FIELD IN DIELECTRICS 

2.1. Polar and Non-Polar Molecules 61 

2.2. Polarization of Dielectrics 63 

2.3. The Field Inside a Dielectric 64 

2.4. Space and Surface Bound Charges 65 

2.5. Electric Displacement Vector 70 

2.6. Examples of Calculating the Field in Dielectrics 73 

2.7. Conditions on the Interface Between Two Dielectrics 77 

2.8. Forces Acting on a Charge in a Dielectric 81 

2.9. Ferroelectrics 82 


3. CONDUCTORS IN AN ELECTRIC FIELD 

3.1. Equilibrium of Charges on a Conductor 84 

3.2. A Conductor in an External Electric Field 86 

3.3. Capacitance 87 

3.4. Capacitors 89 

4. ENERGY OF AN ELECTRIC FIELD 

4.1. Energy of a Charged Conductor 92 

4.2. Energy of a Charged Capacitor 92 

4.3. Energy of an Electric Field 95 



Contents 


1 


CHAPTER 5. STEADY ELECTRIC CURRENT 

5.1. Electric Current 98 

5.2. Continuity Equation 100 

5.3. Electromotive Force 102 

5.4. Ohm’s Law. Resistance of Conductors 104 

5.5. Ohm’s Law for an Inhomogeneous Circuit Section 106 

5.6. Multiloop Circuits. Kirchhoff’s Rules 108 

5.7. Power of a Current 110 

5.8. The Joule-Lenz Law 112 

CHAPTER 6. MAGNETIC FIELD IN A VACUUM 

6.1. Interaction of Currents 113 

4- 6.2. Magnetic Field 115 

6.3. Field of a Moving Charge 116 

6.4. The Biot-Savart Law 120 

6.5. The Lorentz Force 122 

6.6. Ampere’s Law 125 

6.7. Magnetism as a Relativistic Effect 127 

6.8. Current Loop in a Magnetic Field 133 

6.9. Magnetic Field of a Current Loop 138 

-f 6.10. Work Done When a Current Moves in a Magnetic Field 140 

6.11. Divergence and Curl of a Magnetic Field 144 

6.12. Field of a Solenoid and Toroid 148 

CHAPTER 7. MAGNETIC FIELD IN A SUBSTANCE 

7.1. Magnetization of a Magnetic 153 

7.2. Magnetic Field Strength 154 

7.3. Calculation of the Field in Magnetics 160 

7.4. Conditions at the Interface of Two Magnetics 162 

7.5. Kinds of Magnetics 166 

7.6. Gyromagnetic Phenomena 166 

7.7. Diamagnetism 171 

7.8. Paramagnetism 174 

4 - 7.9. Ferromagnetism 177 

CHAPTER 8. ELECTROMAGNETIC INDUCTION 

8.1. The Phenomenon of Electromagnetic Induction 182 

8.2. Induced E.M.F. 183 

8.3. Ways of Measuring the Magnetic Induction 187 

8.4. Eddy Currents 188 

8.5. Self-Induction 189 

8.6. Current When a Circuit Is Opened or Closed 192 

8.7. Mutual Induction 194 

8.8. Energy of a Magnetic Field 196 

8.9. Work in Magnetic Reversal of a Ferromagnetic 198 

CHAPTER 9. MAXWELL’S EQUATIONS 

- 1 9.1. Vortex Electric Field 200 

9.2. Displacement Current 202 
'Jr 9.3. Maxwell’s Equations 206 

CHAPTER 10. MOTION OF CHARGED PARTICLES IN ELECTRIC AND MAGNETIC 
FIELDS 

10.1. Motion of a Charged Particle in a Homogeneous Mag- 
netic Field 209 



8 


Contents 


10.2. Deflection of Moving Charged Particles by an Electric 
and a Magnetic Field 210 

10.3. Determination of the Charge and Mass of an Electron 214 

10.4. Determination of the Specific Charge of Ions. Mass 
Spectrographs 219 

10.5. Charged Particle Accelerators 223 

CHAPTER 11. THE CLASSICAL THEORY OF ELECTRICAL CONDUCTANCE OF 
METALS 

11.1. The Nature of Current Carriers in Metals 228 

11.2. The Elementary Classical Theory of Metals 230 

11.3. The Hall Effect 234 

CHAPTER 12. ELECTRIC CURRENT IN GASES 

12.1. Semi-Self-Sustained and Self-Sustained Conduction 237. 

12.2. Semi-Self-Sustained Gas Discharge 237 

12.3. Ionization Chambers and Counters 240 

12.4. Processes Leading to the Appearance of Current Carriers 
in a Self-Sustained Discharge 245 

12.5. Gas-Discharge Plasma 249 

12.6. Glow Discharge 251 

12.7. Arc Discharge 254 

12.8. Spark and Corona Discharges 255 

CHAPTER 13. ELECTRICAL OSCILLATIONS 

13.1. Quasistationary Currents 259 

13.2. Free Oscillations in a Circuit Without a Resistance 259 

13.3. Free Damped Oscillations 263 

13.4. Forced Electrical Oscillations 266 

13.5. Alternating Current 271 


PART II. WAVES 275 

CHAPTER 14. ELASTIC WAVES 

14.1. Propagation of Waves in an Elastic Medium 275 

14.2. Equations of a Plane and a Spherical Wave 278 

14.3. Equation of a Plane Wave Propagating in an Arbitrary 
Direction 281 

14.4. The Wave Equation 283 

14.5. Velocity of Elastic Waves in a Solid Medium 284 

14.6. Energy of an Elastic Wave 286 

14.7. Standing Waves 290 

14.8. Oscillations of a String 293 

14.9. Sound 294 

14.10. The Velocity of Sound in Gases 297 

14.11. The Doppler Effect for Sound Waves .302 

CHAPTER 15. ELECTROMAGNETIC WAVES 

__ 15.1. The Wave Equation for an Electromagnetic Field 304 
i 15.2. Plane Electromagnetic Wave 305 

15.3. Experimental Investigation of Electromagnetic Waves 309 



Contents 


15.4. Energy of Electromagnetic Waves 310 

15.5. Momentum of Electromagnetic Field 312 

15.6. Dipole Emission 314 


PART III. OPTICS 318 

CHAPTER 16. GENERAL 

16.1. The Light Wave 318 

46 .S r . Representation of Harmonic Functions Using Expo- 
nents 321 

f6v3; Reflection and Refraction of a Plane Wave at the 
Interface Between Two Dielectrics 323 

16.4. Luminous Flux 328 
16:5“. Photometric Quantities and Units 330 
464* Geometrical Optics 333 

16.7. Centered Optical System 337 

16.8. Thin Lens 344 

16.9. Huygens’ Principle 346 

CHAPTER 17. INTERFERENCE OF LIGHT 

17.1. Interference of Light Waves 348 

17.2. Coherence 353 

17.3. Ways of Observing the Interference of Light 361 

17.4. Interference of Light Reflected from Thin Plates 363 
17 t 5'7 The Michelson Interferometer 373 

-17.6.- Multibeam Interference 376 

CHAPTER 18. DIFFRACTION OF LIGHT 

48.1. Introduction 384 

18.2. Huygens-Fresnel Principle 385 

18.3. Fresnel Zones 387 

18.4. Fresnel Diffraction from Simple Barriers 392 

18.5. Fraunhofer Diffraction from a Slit 403 

18.6. Diffraction Grating 411 
18.7; Diffraction of X-Rays 419 

18.8. Resolving Power of an Objective 425 

18.9. Holography 427 

CHAPTER 19. POLARIZATION OF LIGHT 

19.1. Natural and Polarized Light 432 

19.2. Polarization in Reflection and Refraction 436 

19.3. Polarization in Double Refraction 440 

19.4. Interference of Polarized Rays 443 

49.5. Passing of Plane-Polarized Light Through a Crystal 
Plate 445 

19.6. A Crystal Plate Between Two Polarizers 447 

19.7. Artificial Double Refraction 451 

19.8. Rotation of Polarization Plane 453 

CHAPTER 20. INTERACTION OF ELECTROMAGNETIC WAVES WITH A SUBSTANCE 

20.1. Dispersion of Light 456 
Group Velocity 456 



10 


Contents 


20.3. Elementary Theory of Dispersion 462 

20.4. Absorption of Light 466 

20.5. Scattering of Light 467 

20.6. The Vavilov-Cerenkov Effect 470 

XHAPTER MOVING-MEDIA OPTICS 

21.1. The Speed of Light 472 

21.2. Fizeau’s Experiment 474 

21.3. Michelson’s Experiment 477 

21.4. The Doppler Effect 481 

APPENDICES 485 

A.l. List of Symbols 485 

A. 2. Units of Electrical and Magnetic Quantities in the Inter- 
national System (SI) and in the Gaussian System 487 
A. 3. Basic Formulas of Electricity and Magnetism in the SI 
and in the Gaussian System 489 

Name Index 495 
Subject Index 497 




PART I ELECTRICITY 

AND MAGNETISM 


CHAPTER 1 ELECTRIC FIELD 

IN A VACUUM 


1.1. Electric Charge 

All bodies in nature are capable of becoming electrified, i.e. acquir- 
ing an electric charge. The presence of such a charge manifests 
itself in that a charged body interacts with other charged bodies. 
Two kinds of electric charges exist. They are conventionally called 
positive and negative. Like charges repel each other, and unlike 
charges attract each other. 

An electric charge is an integral part of certain elementary parti- 
cles*. The charge of all elementary particles (if it is not absent) is 
identical in magnitude. It can be called an elementary charge. 
We shall use the symbol e to denote a positive elementary charge. 

The elementary particles include, in particular, the electron 
(carrying the negative charge — e), the proton (carrying the positive 
charge +e), and the neutron (carrying no charge). These particles 
are the bricks which the atoms and molecules of any substance are 
built of, therefore all bodies contain electric charges. The particles 
carrying charges of different signs are usually present in a body in 
equal numbers and are distributed over it with the same density. 
The algebraic sum of the charges in any elementary volume of the 
body equals zero in this case, and each such volume (as well as the 
body as a whole) will be neutral. If in some way or other we create 
a surplus of particles of one sign in a body (and, correspondingly, 
a shortage of particles of the opposite sign), the body will be charged. 
It is also possible, without changing the total number of positive 
and negative particles, to cause them to be redistributed in a body 
so that one part of it has a surplus of charges of one sign and the 


♦ Elementary particles are defined as such microparticles whose internal 
structure at the present level of development of physics cannot be conceived as 
a combination of other particles. 



12 


Electricity and Magnetism 


other part a surplus of charges of the opposite sign. This can b© 
done by bringing a charged body close to an uncharged metal one. 

Since a charge q is formed by a plurality of elementary charges, 
it is an integral multiple of e: 

q = ±Ne (1.1) 

An elementary charge is so small, however, that macroscopic charges 
may be considered to have continuously changing magnitudes. 

If a physical quantity can take on only definite discrete values, 
it is said to be quantized. The fact expressed by Eq. (1.1) signifies 
that an electric charge is quantized. 

The magnitude of a charge measured in different inertial reference 
frames will be found to be the same. Hence, an electric charge is 
relativistically invariant. It thus follows that the magnitude of 
a charge does not depend on whether the charge is moving or at rest. 

Electric charges can vanish and appear again. Two elementary 
charges of opposite signs always appear or vanish simultaneously, 
however. For example, an electron and a positron (a positive elec- 
tron) meeting each other annihilate, i.e. transform into neutral 
gamma-photons. This is attended by vanishing of the charges — e 
and -j-e. In the course of the process called the birth of a pair, a gam- 
ma-photon getting into the field of an atomic nucleus transforms 
into a pair of particles — an electron and a positron. This process 
causes the charges — e and +£ to appear. 

Thus, the total charge of an electrically isolated system * cannot 
change . This statement forms the law of electric charge conser- 
vation. 

We must note that the law of electric charge conservation is asso- 
ciated very closely with the relativistic invariance of a charge. 
Indeed, if the magnitude of a charge depended on its velocity, then 
by bringing charges of one sign into motion we would change the 
total charge of the relevant isolated system. 


1.2. Coulomb’s Law 

The law obeyed by the force of interaction of point charges was 
established experimentally in 1785 by the French physicist Charles 
A. de Coulomb (1736-1806). A point charge is defined as a charged 
body whose dimensions may be disregarded in comparison with 
the distances from this body to other bodies carrying an electric 
charge. 

Using a torsion balance (Fig. 1.1) similar to that employed by 
H. Cavendish to determine the gravitational constant (see Vol. I, 

♦ A system is referred to as electrically isolated if no charged particles can 
penetrate through the surface confining it. 



Electric Field in a Vacuum 


13 


p. 174), Coulomb measured the force of interaction of two charged 
spheres depending on the magnitude of the charges on them and 
on the distance between them. He proceeded from the fact that 
when a charged metal sphere was touched by an identical uncharged 
sphere, the charge would be distributed equally be- 
tween the two spheres. 

As a result of his experiments, Coulomb arrived at 
the conclusion that the force of interaction between 
two stationary point charges is proportional to the 
magnitude of each of them and inversely proportional 
to the square of the distance between them . The direc- 
tion of the force coincides with the straight line 
connecting the charges. 

It must be noted that the direction of the force 
of interaction along the straight line, connecting the 
point charges follows from considerations of sym- 
metry. An empty space is assumed to be homogeneous 
and isotropic. Consequently, the only direction dis- 
tinguished in the space by stationary point charges 
introduced into it is that from one charge to the 
other. Assume that the force F acting on the charge 
9i (Fig- 1-2) makes the angle a with the direction 
from q x to g 2 , and that a differs from 0 or ji. But owing to axial 
symmetry, there are no grounds to set the force F aside from the 
multitude of forces of other directions making the same angle a with 



Fig. 1.1 



Fig. 1.2 


Fig. 1.3 


the axis q 1 -q 2 (the directions of these forces form a cone with a cone 
angle of 2a). The difficulty appearing as a result of this vanishes 
when a equals 0 or n. 

Coulomb's law can be expressed by the formula 


F 12 = 




-12 


( 1 . 2 ) 


Here k = proportionality constant assumed to be positive 
q x and q t = magnitudes of the interacting charges 
r = distance between the charges 
e lt = unit vector directed from the charge q t to q t 
F lt = force acting on the charge q x (Fig. 1.3; the figure cor 
responds to the case of like charges). 




14 


Electricity and Magnetism 


The force F n differs from F 12 in its sign: 

F 21 = *-Ms-e 12 (1.3) 

The magnitude of the interaction force, which is the same for 

both charges, can be written in the form 

F = k^- (1.4) 

Experiments show that the force of interaction between two given 

charges does not change if other charges are placed near them. Assume 
that we have the charge q a and, in addition, N other charges 
qi, If can b e seen from the above that the resultant 

force F with which all the N charges q x act on q a is 

F= 2 F a ., (1.5) 

1-1 

where F a ,* is the force with which the charge q t acts on g a in the 
absence of the other N — 1 charges. 

The fact expressed by Eq. (1.5) permits us to calculate the force 
of interaction between charges concentrated on bodies of finite 
dimensions, knowing the law of interaction between point charges. 
For this purpose, we must divide each charge into so small charges dq 
that they can be considered as point ones, use Eq. (1.2) to calculate 
the force of interaction between the charges dq taken ijX -pair s, and 
then perform vector summation of these forces. Mathematically, 
this procedure coincides completely with the calculation of the 
force of gravitational attraction between bodies of finite dimensions 
(see Vol. I, Sec. 6.1). 

All experimental facts available lead to the conclusion that 
Coulomb’s law holds for distances from 10“ 15 m to at least several 
kilometres. There are grounds to presume that for distances smaller 
than 10~ 16 m the law stops being correct. For very great distances, 
there are no experimental confirmations of Coulomb’s law. But 
there are also no reasons to expect that this law stops being obeyed 
with very great distances between charges. 


1.3. Systems of Units 

We can make the proportionality constant in Eq. (1.2) equal 
unity by properly choosing the unit of charge (the units for F and r 
were established in mechanics). The relevant unit of charge (when 
F and r are measured in cgs units) is called the absolute electrostatic 
unit of charge (cgse 9 ). It is the magnitude of a charge that interacts 


Electric Field in a Vacuum 


iS 


with a force of 1 dyne in a vacuum with an equal charge at a distance 
of 1 cm from it. 

Careful measurements (they are described in Sec. 10.3) showed 
that an elementary charge is 

e = 4.80 x 10~ ! ° cgse* (1.6) 

Adopting the units of length, mass, time, and charge as the basic 
ones, we can construct a system of units of electrical and magnetic 
quantities. The system based on the centimetre, gramme, second, 
and the cgse* unit is called the absolute electrostatic system of units 
(the cgse system). It is founded on Coulomb’s law, i.e. the law of 
interaction between charges at rest. On a later page, we shall become 
acquainted with the absolute electromagnetic system of units (the 
cgsm system) based on the law of interaction between conductors 
carrying an electric current. The Gaussian system in which the units 
of electrical quantities coincide with those of the cgse system, and 
of magnetic quantities with those of the cgsm system, is also an 
absolute system. 

Equation (1.4) in the cgse system becomes 

(1.7) 

This equation is correct if the charges are in a vacuum. It has to be 
determined more accurately for charges in a medium (see Sec. 2.8). 

USSR State Standard GOST 9867-61, which came into force 
on January 1, 1963. prescribes the preferable use of the Internation- 
al System of Units (SI). The basic units of this system are the 
metre, kilogramme, second, ampere, kelvin, candela, and mole. 
The SI unit of force is the newton (N) equal to 10 5 dynes. 

In establishing the units of electrical and magnetic quantities, 
the SI system, like the cgsm one, proceeds from the law of interac- 
tion of current-carrying conductors instead of charges. Consequently, 
the proportionality constant in the equation of Coulomb’s law is 
a quantity with a dimension and differing from unity. 

The SI unit of charge is the coulomb (C). It has been found experi- 
mentally that 

1 C « 2.998 x 10® « 3 x 10® cgse* (1.8) 

To form an idea of the magnitude of a charge of 1 C, let us calcu- 
late the force with which two point charges of 1 C each would interact 
with each other if they were 1 m apart. By Eq. (1*7) 

F = 3X1 °; . ^X1 . °» cgsef = 9 x 10 « dyn = 9 x 10» N « 10* kgf (1.9) 

An elementary charge expressed in coulombs is 

e= 1.60 x 10~ 1# C 


( 1 . 10 ) 




16 


Electricity and Magnetism 


1.4. Rationalized Form 
of Writing Formulas 


Many formulas of electrodynamics when written in the cgs systems 
(in particular, in the Gaussian one) include as factors 4 n and the 
so-called electromagnetic constant c equal to the speed of light 
in a vacuum. To eliminate these factors in the formulas that are 
most important in practice, the proportionality constant in Cou- 
lomb's law is taken equal to l/4rc6 0 . The equation of the law for 
charges in a vacuum will thus become 




1 1 gl<72 I 

4 ne 0 r 2 


(l.ii) 


The other formulas change accordingly. This modified way of writ- 
ing formulas is called rationalized. Systems of units constructed 
with the use of rationalized formulas are also called rationalized. 
They include the SI system. 

The quantity e 0 is called the electric constant. It has the dimension 
of capacitance divided by length. It is accordingly expressed in 
units called the farad per metre. To find the numerical value of e 0 , 
we shall introduce the values of the quantities corresponding to the 
case of two charges of 1 C each and 1 m apart into Eq. (1.11). By 
Eq. (1.9), the force of interaction in this case is 9 X 10 9 N. Using 
this value of the force, and also q x = q 2 ■ = 1 C and r = 1 m in 
Eq. (1.11), we get 


9 x 10 9 = 


1 

4neo 


IXl 

1* 


whence 




4 n X 9 x 10 * 


= 0.885 x 10~ !1 F/m 


( 1 . 12 ) 


The Gaussian system of units was widely used and is continuing 
to be used in physical publications. We therefore consider it essen- 
tial to acquaint our reader with both the SI and the Gaussian system. 
We shall set out the material in the SI units showing at the same 
time how the formulas look in the Gaussian system. The fundamen- 
tal formulas of electrodynamics written in the SI and the Gaussian 
system are compared in Appendix 3. 


1.5. Electric Field. Field Strength 

Charges at rest interact through an electric field*. A charge alters 
the properties of the space surrounding it — it sets up an electric 
field in it. This field manifests itself in that an electric charge placed 

• We shall see in Sec. 6.2 that when considering moving charges, their inter- 
action in addition to an electric field is due to a magnetic field. 




Electric Field in a Vacuum 


17 


at a point of it experiences the action of a force. Hence, to see whether 
there is an electric field at a given place, we must place a charged 
body (in the following we shall say simply a charge for brevity) at 
it and determine whether or not it experiences the action of an 
electric force. We can evidently assess the “strength” of the field 
according to the magnitude of the 
force exerted on the given charge. 

Thus, to detect and study an electric 
field, we must use a “test” charge. For 
the force acting on our test charge to 
characterize the field “at the given 
point”, the test charge must be a point 
one. Otherwise, the force acting on the 
charge will characterize the properties 
of the field averaged over the volume 
occupied by the body that carries the 
test charge. 

Let us study the field set up by the 
stationary point charge q with the aid 
of the point test charge q t . We place the test charge at a point whose 
position relative to the charge q is etermined by the position vector 
r (Fig. 1.4). We see that the test charge experiences the force 

F -M4^T-3-«') < )i3 > 

[see Eqs. (1.3) and (1.11)]. Here e r is the unit vector of the position 
vector r. 

A glance at Eq. (1.13) shows that the force acting on our test charge 
depends not only on the quantities determining the field (on q and r), 
but also on the magnitude of the test charge q t . If we take different 
test charges q[, q{, etc., then the forces F', F", etc. which they 
experience at the given point of the field will be different. We can 
see from Eq. (1.13), however, that the ratio F/g t for all the test 
charges will be the same and depend only on the values of q and r 
determining the field at the given point. It is therefore natural 
to adopt this ratio as the quantity characterizing an electric field: 

E = — (1.14) 

9t 

This vector quantity is called the electric field strength (or intensity) 
at a given point (i.e. at the point where the test charge q t experi- 
ences the action of the force F). 

According to Eq. (1.14), the electric field strength numerically 
equals the force acting on a unit point charge at the given point 
of the field. The direction of the vector E coincides with that of the 
force acting on a positive charge. 



Fig. 1.4 



18 


Electricity and Magnetism 


It must be noted that Eq. (1.14) also holds when the test charge 
is negative (g t < 0). In this case, the vectors E and F have opposite 
directions. 

We have arrived at the concept of electric field strength when 
studying the field of a stationary point charge. Definition (1.14), 
however, also covers the case of a field set up by any collection of 
stationary charges. But here the following clarification is needed. 
The arrangement of the charges setting up the field being studied 
may change under the action of the test charge. This will happen, 
for example, when the charges producing the field are on a conduc- 
tor and can freely move within its limits. Therefore, to avoid appre- 
ciable alterations in the field being studied, a sufficiently small test 
charge must be taken. 

It follows from Eqs. (1.14) and (1.13) that the field strength of 
a point charge varies directly with the magnitude of the charge q 
and inversely with the square of the distance r from the charge to 
the given point of the field: 


E 


1 q 
4jieo r* 


er 


(1.15) 


The vector E is directed along the radial straight line passing through 
the charge and the given point of the field, from the charge if the 
latter is positive and toward the charge if it is negative. 

In the Gaussian system, the equation for the field strength of 
a point charge in a vacuum has the form 



(1.16) 


The unit of electric field strength is the strength at a point where 
unit force (1 N in the SI and 1 dyn in the Gaussian system) acts 
on unit charge (1 C in the SI and 1 cgse 9 in the Gaussian system). 
This unit has no special name in the Gaussian system. The SI unit 
of electric field strength is called the volt per metre (V/m) [see 
Eq. (1.44)1. 

According to Eq. (1.15), a charge of 1 C produces the following 
field strength in a vacuum at a distance of 1 m from this charge: 


E 4ji (1/4ji x9 X i0*) 1* 9 X 10* V/m 


This strength in the Gaussian system is 


£ = = cgse* 


Comparing these two results, we find that 

1 cgse B = 3 x 10 4 V/m 


(1.17) 




Electric Field in a Vacuum 


19 


According to Eq. (1.14), the force exerted on a test charge is 

F = g t E 

It is obvious that any point charge q* at a point of a field with the 
strength E will experience the force 

F = qE (1.18) 

If the charge q is positive, the direction of the force coincides with 
that of the vector E. If q is negative, the vectors F and E are directed 
oppositely. 

We mentioned in Sec. 1.2 that the force with which a system of 
charges acts on a charge not belonging to the system equals the 
vector sum of the forces which each of the charges of the system 
exerts separately on the given charge [see Eq. (1.5)]. Hence it fol- 
lows that the field strength of a system of charges equals' the vector sum 
of the field strengths that would be produced by each of the charges of 
the system separately : (vy 

E=SE, (1.19) 

This statement is called the principle of electric field superposition* 

The superposition principle allows us to calculate the field strength 
of any system of charges. By dividing extended charges into suffi- 
ciently small fractions dq , we can reduce 
any system of charges to a collection of 
point charges. We calculate the contribu- 
tion of each of such charges to the resul- 
tant field by Eq. (1.15). 

An electric field can be described by 
indicating the magnitude and direction of 
the vector E for each of its points. The 
combination of these vectors forms the 
field of the electric field strength vector 
(compare with the field of the velocity vector, Vol. I, p. 249). The 
velocity vector field can be represented very illustratively with the 
aid of flow lines. Similarly, an electric field can be described with 
the aid of strength lines, which we shall call for short E lines or 
field lines. These lines are drawn so that a tangent to them at every 
point coincides with the direction of the vector E. The density of 
the lines is selected so that their number passing through a unit 
area at right angles to the lines equals the numerical value of the 
vector E. Hence, the pattern of field lines permits us to assess the 
direction and magnitude of the vector E at various points of space 
(Fig. 1.5). 

The E lines of a point charge field are a collection of radial straight 
lines directed away from the charge if it is positive and toward it if 

* In Eq. (1.15), q stands for the charge setting up the field. In Eq. (1.18), 
q stands for the charge experiencing the force F at a point of strength E. 




20 


Electricity and Magnetism 


it is negative (Fig. 1.6). One end of each line is at the charge, and 
the other extends to infinity. Indeed, the total number of lines 
'intersecting a spherical surface of arbitrary radius r will equal the 
product of the density of the lines and the surface area of the sphere 
4nr 2 . We have assumed that the density of the lines numerically 
equals E = (l/4jie 0 ) (q/r 2 ). Hence, the number of lines is 
(l/4ne 0 ) (q/r 2 ) Anr 2 = q/e 0 . This result signifies that the number of 
lines at any distance from a charge will be the same. It thus follows 



fhat the lines do not begin and do not terminate anywhere except 
tor the charge. Beginning at the charge, they extend to infinity (the 
charge is positive), or arriving from infinity, they terminate at thrf 
charge (the latter is negative). This property of the E lines is com- 
mon for all electrostatic fields, i.e. fields set up by any system of 
stationary charges: the field lines can begin or terminate only at 
charges or extend to infinity. 


1.6. Potential 


Let us consider the field produced by a stationary point charge q. 
At any point of this field, the point charge q' experiences the force t 


F = 


1 W 

4ne 0 r 2 


e r = F (r) e r 


( 1 . 20 )' 


Here F (r) is the magnitude of the force F, and e r is the unit vector 
of the position vector r determining the position of the charge q' 
relative to the charge q. 

The force (1.20) is a central one (see Vol. I, p. 82). A central 
field of forces is conservative. Consequently, the work done by the 
forces of the field on the charge q' when it is moved from one point 


Electric Field in a Vacuum 


21 


to another does not depend on the path. This work is 


2 


A i2 



e r d\ 


( 1 . 21 ) 


where d\ is the elementary displacement of the charge q . Inspection 
of Fig. 1.7 shows that the scalar product e r d\ equals the increment 
of the magnitude of the position vector r, i.e. dr. Equation (1.21) 
can therefore be written in the form 
2 

^ 12 = f F(r)dr 
1 

[compare with Eq. (3.24) of Vol. I, p. 84]. In- 
troduction of the expression for F (r) yields 
. r * 


M2 - 


qq 


4ne 0 


f dr __ 1 / qq’ qq' \ 

J r 1 4nco \ r x r 2 ) 


r x r 2 

( 1 . 22 ) 

The work of the forces of a conservative 
field can be represented as a decrement of 
the potential energy: 



M2 * 


W Pti -W 


P. 2 


(1.23) 


Pig. 1.7 


A comparison of Eqs. (1.22) and (1.23) leads to the following expres- 
sion for the potential energy of the charge q’ in the field of the 
charge q: 

1 99 ' 


^ = '4^ 


+ const 


The value of the constant in the expression for the potential energy 
is usually chosen so that when the charge moves away to infinity 
(i.e. when r = oo), the potential energy vanishes. When this con- 
dition is observed, we get 


W* 


1 qq' 

4neo r 


(1.24) 


Let us use the charge q ' as a test charge for studying the field. 
By Eq. (1.24), the potential energy which the test charge has depends 
not only on its magnitude q\ but also on the quantities q and r 
determining the field. Thus, we can use this energy to describe the 
field just like we used the force acting on the test charge for this 
purpose. 

Different test charges q \ , q\, etc. will have different energies 
IFp, W pi etc. at the same point of a field. But the ratio W p /q t will 
be the same for all the charges [see Eq. (1.24)]. The quantity 

W p 



22 


Electricity and Magnetism 


is called the field potential at a given point and is used together with 
the field strength E to describe electric fields. 

It can be seen from Eq. (1.25) that the potential numerically 
equals the potential energy which a unit positive charge would have 
at the given point of the field. Substituting for the potential energy 
in Eq. (1.25) its value from (1.24), we get the following expression 
for the potential of a point charge: 

c- 26 ) 

In the Gaussian system, the potential of the field of a point charge 
in a vacuum is determined by the formula 

q>=j- (1.27) 


Let us consider the field produced by a system of N point charges 
<h<> £ 2 * - • -» Qn • Let r lt r 2 , . . r N be the distances from each 
of the charges to the given point of the field. The work done by the 
forces of this field on the charge q ' will equal the algebraic sum of the 
work done by the forces set up by each of the charges separately: 

i=i 

By Eq, (1.22), each work A t equals 

a 1 / w' w' \ 

1 4 n% V r*. j r iy% ) 


where r i% x is the distance from the charge q t to the initial position 
of the charge q\ and r i% 2 is the distance from q t to the final position 
of the charge g'. Hence, 

n ” 

A 1 V qiq ' 1 V qtqf 

12 4jieo ^ ri, ! 4neo " n t t 

i=l i=l 


Comparing this equation with Eq. (1.23), we get the following 
expression for the potential energy of the charge q' in the field of 
a system of charges: 



N 


Is 


g!9 f 

n 


from which it can be seen that 

^ ~ 4neo 


N 





(1.28) 



Electric Field in a Vacuum 


23 


Comparing this formula with Eq. (1.26)* we arrive at the conclu- 
sion that the potential of the field produced by a system of charges 
equals the algebraic sum of the potentials produced by each of the charges 
separately . Whereas the field strengths are added vectorially in the 
superposition of fields* the potentials are added algebraically. This 
is why it is usually much simpler to calculate the potentials than 
the electric field strengths. 

Examination of Eq. (1.25) shows that the charge q at a point 
of a field with the potential <p has the potential energy 

W* - qq> (1.29) 

Hence, the work of the field forces on the charge q can be expressed 

through the potential difference: 

Ai 2 = W P ' t -W p . x =q "<Pi — SPi) (1-30) 

Thus* the work done on a charge by the forces of a field equals the 
product of the magnitude of the charge and the difference between 
the potentials at the initial and final points (i.e. the potential decre- 
ment). 

If the charge q is removed from a point having the potential <p 
to infinity (where by convention the potential vanishes), then the 
work of the field forces will be 

A oo = q<p (1*31) 

He e, it follows that the potential numerically equals the work done 
by the forces of a field on a unit positive charge when the latter is removed 
from the given point to infinity . Work of the same magnitude must 
be done against the electric field forces to move a unit positive charge 
from infinity to the given point of a field. 

Equation (1.31) can be used to establish the units of potential. 
The unit of potential is taken equal to the potential at a point 
of a field when work equal to unity is required to move unit positive 
charge from infinity to this point. The SI unit of potential called 
the volt (V) is taken equal to the potential at a point when work 
of 1 joule has to be done to move a charge of 1 coulomb from infinity 
to this point: 

H=lCxlV 

whence 

1 V = (1.32) 

The absolute electrostatic unit of potential (cgse 9 ) is taken equal 
to the potential at a point when work of 1 erg has to be done to move 
a charge of +1 cgse 9 from infinity to this point. Expressing 1 J 
and 1 C in Eq. (1.32) through cgse units, we shall find the relation 



24 


Electricity and Magnetism 


between the volt and the cgse potential unit: 

1 V = T = (1 ' 33) 

Thus, 1 cgse,, equals 300 V. 

A unit of energy and work called the electron-volt (eV) is frequent- 
ly used in physics. An electron-volt is defined as the work done by 
the forces of a field on a charge equal to that of an electron (i.e. on 
the elementary charge e) when it passes through a potential dif- 
ference of 1 V: 

1 eV = 1.60 x 10~ 19 Cxi V = 1.60 X 10 -19 J = 1.60 x 10^ erg 

(1.34) 

Multiple units of the electron-volt are also used: 

1 keV (kiloelectron-volt) = 10 s eV 
1 MeV (megaelectron-volt) — 10 s eV 
1 GeV (gigaelectron-volt) = 10® eV 


1.7. Interaction Energy 
of a System of Charges 

Equation (1.24) can be considered as the mutual potential energy 
of the charges q and q\ Using the symbols q 2 and q 2 for these charges, 
we get the following formula for their interaction energy: 

< 13S > 

The symbol r 12 stands for the distance between the charges. 

Let us consider a system consisting of N point charges q u q 2 , . . . 
. . q N . We showed in Sec. 3.6 of Vol. I (p. 95) that the energy of 

interaction of such a system equals the sum of the energies of interac- 
tion of the charges taken in pairs: 

(1.36) 


(1.37) 




(i ¥*) 


{see Eq. (3.60) of Vol. I, p.95]. 
According to Eq. (1.35) 


W 


p. th - 


orn 


4neo rih 

Using this equation in (1.36), we And that 

w —— V l Mi. 

P 2 4neo rj* 




Electric Field in a Vacuum 


25 


In the Gaussian system, the factor l/4jie 0 is absent in this equa- 
tion. 

In Eq. (1.37), summation is performed over the subscripts i and k. 
Both subscripts pass independently through all the values from 1 
to N. Addends for which the value of the subscript i coincides with 
that of k are not taken into consideration. Let us write Eq. (1.37) 
as follows: 


N N 

i=l ft=l 

(fe^O 


_J 9*. 

4«e 0 rjft 


(1.38) 


The expression 


<Pi 


N 

1 y J7*_ 
4ne 0 Zj r ifc 
A — 1 


is the potential produced by all the charges except q t at the point 
where the charge q t is. With this in view, we get the following 
formula for the interaction energy: 


N 

^ = 4-2 Mi 

1=1 


(1.39) 


1.8. Relation Between Electric Field 
Strength and Potential 

An electric field can be described either with the aid of the vector 
quantity E, or with the aid of the scalar quantity <p. There must 
evidently be a definite relation between these quantities. If we 
bear in mind that E is proportional to the force acting on a charge 
and <p to the potential energy of the charge, it is easy to see that 
this relation must be similar to that between the potential energy 
and the force. 

The force F is related to the potential energy by the expression 

F = —vIFp (1.40) 

[see Eq. (3.32) of Vol. I, p. 88]. For a charged particle in an electro- 
static field, we have F = ^E and W p = qt p. Introducing these values 
into Eq. (1.40), we find that 

qE = —V (q<f) 

The constant q can be put outside the gradient sign* Doing this 
and then cancelling q 7 we arrive at the formula 

E = ~v<p ( 1 - 41 ) 

establishing the relation between the field strength and potential. 


26 


Electricity and Magnetism 


Taking into account the definition of the gradient [see Eq. (3,31) 
of Vol. 1, p. 88], we can write that 


E = 


dy 

dx 



(1.42) 


Hence, Eq. (1.41) has the following form in projections onto the 
coordinate axes: 


Ex 


a<p p j><p_ p ^9 

dx * dy ’ I_ 


(1.43) 


Similarly, the projection of the vector E onto an arbitrary direc- 
tion l equals the derivative of 9 with respect to l taken with the 
opposite sign, i.e. the rate of diminishing of the potential when 
moving along the direction l : 

(1.44) 

It is easy to see that Eq. (1.44) is correct by choosing l as one of the 
coordinate axes and taking Eq. (1.43) into account. 

Let us explain Eq. (1.41) using as an example the field of a point 
charge. The potential of this field is expressed by Eq. (1.26). Passing 
over to Cartesian coordinates, we get the expression 

m = * !L — — 1 g 

^ 4 ne 0 r 4ne 0 y'' x 2 + y 2 +z t 

The partial derivative of this function with respect to x is 

1 q* — l gf 

4nco (x* + y*+**) 3/2 4 jw^ H 

1 qy dy 1 q% 

— 4neq r* 9 dz 4neo r* 

Using the found values of the derivatives in Eq (1.42) t we arrive 
at the expression 

F 1 q(xe x +ye v + ze z ) __ 1 qr __ 1 q 

AtiIq r* 4jieo r* ineo r* r 


Similarly 


dtp __ 
dx ~ 


dy 

dy 


that coincides with Eq. (1.15). 

Equation (1.41) allows us to find the field strength at every point 
from the known values of q>. We can also solve the reverse problem, 
i.e. find the potential difference between two arbitrary points of 
a field according to the given values of E. For this purpose, we shall 
take advantage of the circumstance that the work done by the forces 
of a field on the charge q when it is moved from point 1 to point 2 



Electric Field in a Vacuum 


27 


can be calculated as 

2 

A 12 = | qE dl 

At the same time in accordance with Eq. (1.30), this work can be 
written as 

■^12 = 9 (<Pi — <Pi) 

Equating these two expressions and cancelling q, we obtain 

2 

9i-q>2=jEdl (1.45) 

1 

The integral can be taken along any line joining points 1 and 2 
because the work of the field forces is independent of the path. For 
circumvention along a closed contour, <p x = <j> 2 , and Eq. (1.45) 
becomes 

§ E dl = 0 (1.46) 

(the circle on the integral sign indicates that integration is per- 
formed over a closed contour). It must be noted that this relation holds 
only for an electrostatic field. We shall see on a later page that the 
field of moving charges (i.e. a field changing with time) is not a po- 
tential one. Therefore, condition (1.46) is not observed for it. 

An imaginary surface all of whose points have the same potential 
is called an equipotential surface. Its equation has the form 

cp Vi 2 ) = const 

The potential does not change in movement along an equipotential 
surface over the distance dl (d<p = 0). Hence, according to Eq. (1.44), 
the tangential component of the vector E to the surface equals zero. 
We thus conclude that the vector E at every point is directed along 
a normal to the equipotential surface passing through the given 
point. Bearing in mind that the vector E is directed along a tangent 
to an E line, we can easily see that the field lines at every point are 
orthogonal to the equipotential surfaces. 

An equipotential surface can be drawn through any point of 
a field. Consequently, we can construct an infinitely great number of 
such surfaces. They are conventionally drawn so that the potential 
difference for two adjacent surfaces is the same everywhere. Thus, 
the density of the equipotential surfaces allows us to assess the 
magnitude of the field strength. Indeed, the denser are the equipo- 
tential surfaces, the more rapidly does the potential change when 
moving along a normal to the surface. Hence, V<p is greater at the 
given place, and, therefore, E is greater too. 


28 


Electricity and Magnetism 


Figure 1.8 shows equipotential surfaces (more exactly, their 
intersections with the plane of the drawing) for the field of a point 
charge. In accordance with the nature of the dependence of E on r, 
equipotential surfaces become the denser, the nearer we approach 
a charge. 

Equipotential surfaces for a homogeneous field are a collection of 
equispaced planes at right angles to the direction of the field. 


1.9. Dipole 

An electric dipole is defined as a system of two point charges 
+q and — q identical in value and opposite in sign, the distance 


E 




Fig. 1.8 Fig. 1.9 

between which is much smaller than that to the points at which the 
field of the system is being determined. The straight line passing 
through both charges is called the dipole axis. 

Let us first calculate the potential and then the field strength of 
a dipole. This field has axial symmetry. Therefore, the pattern of 
the field in any plane passing through the dipole axis will be the 
same, the vector E being in this plane. The position of a point relative 
to the dipole will be characterized with the aid of the position vec- 
tor r or with the aid of the polar coordinates r and 0 (Fig. 1.9). 
We shall introduce the vector 1 passing from the negative charge 
to the positive one. The position of the charge -f -q relative to the 
centre of the dipole is determined by the vector a, and of the charge 
— q by the vector — a. It is obvious that 1 = 2a. We shall desig- 




Electric Field in a Vacuum 


29 


nate the distances to a given point from the charges +q and — q 
by r+ and r_, respectively. 

Owing to the smallness of a in comparison with r, we can assume 
approximately that 


r+ = r — a cos 0 = r — ae r 
r_ = r + a cos 0 = r + ae r 


(1.47) 


The potential at a point determined by the position vector r is 


< * , ( r ) = 4lk 



1 g(r~— r») 

4 ne 0 r + r_ 


The product r + r__ can be replaced with r 2 . The difference r_ — r + , 
according to Eqs. (1.47), is 2ae r = le r . Hence, 


where 


<p(r) = 


1 gfer 

4neo r s 


1 P£r 

4neo r 2 


(1.48) 


P = ?1 


(1.49) 


is a characteristic of a dipole called its electric moment. The vec- 
tor p is directed along the dipole axis from the negative charge 
to the positive one (Fig. 1.10). 

A glance at Eq. (1.48) shows that the field of _ q +q 

a dipole is determined by its electric moment —9 ■ » p 

p. We shall see below that the behaviour of a * v 
dipole in an external electric field is also deter- * 

mined by its electric moment p. A comparison Fig. 1.10 

with Eq. (1.26) shows that the potential of 

a dipole field diminishes with the distance more rapidly (as 1 /r 2 ) 
than the potential of a point charge field (which changes in propor- 
tion to 1 /r). 

It can be seen from Fig. 1.9 that pe r = p cos 0. Therefore, Eq. (1.48) 
can be written as follows: 


<P (r, 6) = 


1 

4ji6o 


pc os 0 

7* 


(1.50) 


To find the field strength of a dipole, let us calculate the projec- 
tions of the vector E onto two mutually perpendicular directions by 
Eq. (1.44). One of them is determined by the motion of a point due 
to the change in the distance r (with 0 fixed), the other by the mo- 
tion of the point due to the change in the angle 0 (with r fixed, 
see Fig. 1.9). The first projection is obtained by differentiation of 
Eq. (1.50) with respect to r: 


0 <p 12 p cos 0 

dr 4jiso r 3 


(1.51) 


We shall find the second projection (let us designate it by 2?e) hy 
taking the ratio of the increment of the potential <p obtained when 



30 


Electricity and Magnetism 


the angle 0 grows by <20 to the distance r <20 over which the end 
of the segment r moves tin this case the quantity <22 in Eq. (1.44) 
equals r <20 ). Thus, 

jp _ dtp _ 1 d<p 

£ ' e ~ rd0 r d0 


Introducing the value of the derivative of function (1.50) with 
respect to 0, we get 


p 1 p sin 0 
£e== 45Teo'~" 


(1.52) 


The sum of the squares of Eqs. (1.51) and (1.52) gives the square of 
the vector E (see Fig. 1.9): 

Bf - Bi + Ei - ( )’ ( ■£■ f (4 cos* e + sin* 6) = 

= (- sk ) ! (^) 2 ( 1 + 3 “ s8e > 

Hence 

£ -w£^ 1 + 3cos,e < 133 > 


Assuming in Eq. (1.53) that 0 = 0, we get the strength on the 
dipole axis: 


Ei 


1 2 p 

4jteo r* 


(1.54) 


The vector Ey is directed along the dipole axis. This is in agreement 
with the axial symmetry of the problem. Examination of Eq. (1.51) 
shows that E r > 0 when 0 = 0, and E r <L 0 when 0 = n. This 
signifies that in any case the vector E|| has a direction coinciding 
with that from — q to +q (i.e. with the direction of p). Equation 
(1.54) can therefore be written in the vector form: 


E| 


1 2p 

4jicq r* 


(1.55) 


Assuming in Eq. (1.53) that 0 = ji/ 2, we get the field strength 
on the straight line passing through the centre of the dipole and 
perpendicular to its axis: 


F L_ P 


(1.56) 


By Eq. (1.51), when 0 = n/2, the projection E r equals zero. Hence, 
the vector E x is parallel to the dipole axis. It follows from Eq. (1.52) 
that when 0 = n/2, the projection 2?e is positive. This signifies that 
the vector E x is directed toward the growth of the angle 0, i.e. anti- 
parallel to the vector p. 

The field strength of a dipole is characterized by the circumstance 
that it diminishes with the distance from the dipole in proportion 


Electric Field in a Vacuum 


31 


to 1/r 3 , i.e. more rapidly than the field strength of a point charge 
(which diminishes in proportion to 1 /r*). 

Figure 1.11 shows E lines (the solid lines) and equipotential sur- 
faces (the dash lines) of the field of a dipole. According to Eq. (1.50), 
when 0 = n/2, the potential vanishes for all the r’s. Thus, all the 
points of a plane at right angles to the dipole axis and passing 
through its middle have a zero potential. This could have been pre- 
dicted because the distances from the charges -f q and — q to any 
point of this plane are identical. 

Now let us turn to the behaviour of a dipole in an external elec- 
tric field. If a dipole is placed in a homogeneous electric field, the 




Fig. 1.11 Fig. 1.12 

charges +? and — q forming the dipole will be under the action of 
the forces F x and F 2 equal in magnitude, but opposite in direction 
(Fig. 1.12). These forces form a couple whose arm is l sin a, i.e. 
depends on the orientation of the dipole relative to the field. The 
magnitude of each of the forces is qE. Multiplying it by the arm, 
we get the magnitude of thfi-_torque acting on a dipole: 

T — qEl sin a = pE sin a (1*57) 

(p is the electric moment of the dipole). It is easy to see that Eq. (1.57) 
can be written in the. vector form 

T = [pE] (1.58) 

The torque (1.58) tends to turn a dipole so that its electric moment p 
is in the direction of the field. 

Let us find the potential energy belonging to a dipole in an exter- 
nal electric field. By Eq. (1.29), this energy is 

w v = 7<P+ — ?<P- = ? (<P+ — <P-) (1-59) 

Here q> + and <p_ are the values of the potential of the external field 
at the points where the charges 4 -q and — q are placed. 


32 


Electricity and Magnetism 


The potential of a homogeneous field diminishes linearly in the 
direction of the vector E. Assuming that the x-axis is this direction 
(Fig. 1.13), we can write that E = E x = — dy/dx. A glance at 



Fig. 1.13 Fig. 1.14 


Fig. 1.13 shows that the difference <p+ — equals the increment 
of the potential on the segment A x — l cos a: 

<p+ — <p_ — 1 cos a = — El cos a 

Introducing this value into Eq. (1.59), we find that 

W p = — qEl cos a — — pE cos a (1.60) 

Here a is the angle between the vectors p and E. We can therefore 
write Eq. (1.60) in the form 

W p = —pE (1.61) 

We must note that this expression takes no account of the energy 
of interaction of the charges and — q forming a dipole. 

We have obtained Eq. (1.61) assuming for simplicity’s sake that 
the field is homogeneous. This equation also holds, however, for 
an inhomogeneous field. 

Let us consider a dipole in an inhomogeneous field that is sym- 
metrical relative to the x-axis*. Let the centre of the dipole be on 
this axis, the dipole electric moment making with the axis an angle a, 
differing from n/2 (Fig. 1.14). In this case, the forces acting on the 
dipole charges are not identical in magnitude. Therefore, apart from 
the rotational moment (torque), the dipole will experience a force 
tending to move it in the direction of the x-axis. To find the value 
of this force, we shall use Eq. (1.40), according to which 
F x = —dWp/dx, F y = —dW p /dy 7 F z = -OWptdz 
In view of Eq. (1.60), we can write 

Wp (x, y , z) = — pE (x, y, z) cos a 

• A particular case of such a field is that of a point charge if we take a straight 
line passing through the charge as the x-axis. 



Electric Field in a Vacuum 


33 


(we consider the orientation of the dipole relative to the vector E 
to be constant, a = const). 

For points on the x-axis, the derivatives of E with respect to y 
and z are zero. Accordingly, dW p /dy = dW p /dz = 0. Thus, only 
the force component F x differs from zero. It is 

dW v QE 

— jt=p-w cosa ( 162 ) 



This result can be obtained if we take account of the fact that the 
field strength at the points where the charges +q and — q are (see 
Fig. 1.14) differs by the amount 
(dE/dx) Zcos a. Accordingly, the differ- 
ence between the forces acting on the 
charges is q (dE/dx) l cos a, which coin- 
cides with Eq. (1.62). 

When a is less than n/2, the value of 
F x determined by Eq. (1.62) is positive. 

This signifies that under the action 
of the force the dipole is pulled into 
the region of a stronger field (see Fig. 

1.14). When a is greater than ji/ 2, the 
dipole is pushed out of the field. 

In the case shown in Fig. 1.15, only the derivative dE/dy differs 
from zero for points on the y- axis. Therefore, the force acting on the 
dipole is determined by the component 


Fig. 1.15 


F v = 


aw* 


dy 


dE 

dy 


(cos a = 1) 


The derivative dE/dy is negative. Consequently, the force is directed 
as shown in the figure. Thus, in this case too, the dipole is pulled 
into the field. 

We shall note that like — dW p ldx gives the projection of the force 
acting on the system onto the x-axis, so does the derivative of 
Eq. (1.60) with respect to a taken with the opposite sign give the 
projection of the torque onto the a- ‘axis”: T a — — pE sin a. 

The minus sign was obtained because the a-“axis” and the torque T 
are directed oppositely (see Fig. 1.12). 


1.10. Field of a System of Charges at 
Great Distances 

Let us take a system of N charges q ly q 2 , . . ., q N in a volume 
having linear dimensions of the order of Z, and study the field set 
up by this system at distances r that are great in comparison with 
Z (r » Z). We take the origin of coordinates O inside the volume 



34 


Electricity and MagnetUm 


occupied by the system and shall determine the positions of the 
charges with the aid of the position vectors r t (Fig. 1.16; to simplify 

the figure, we have shown only the 



position vector of the i-th charge). 

The potential at the point de- 
termined by the position vector 
r is 

<‘ 63 > 

1=1 

Owing to the smallness of r £ in 
comparison with r, we can as- 
sume that 


Fig - 1 - 16 jr — r,| = r— r,e r = r(l— 

[compare with Eqs. (1.47)]. Introduction of this expression into 
Eq. (1.63) yields 

* w-ts; 2 <‘« 4 > 

t=l 

Using the formula 


which holds when «< 1, we can transform Eq. (1.64) as follows: 


2 T -( 1 + 


fie r 

r 



i 2?* . i (Sg <r O e,> 

4neo r 4neo r* 


(1.65) 


The first term of the expression obtained is the potential of the 
field of a point charge having the value q — 2 q t [compare with 
Eq. (1.26)1. The second term has the same form as the expression 
determining the potential of a dipole field, the part of the electric 
moment of the dipole being played by the quantity 


P 


N 

2 

i=i 


( 1 . 66 ) 


This quantity is called the dipole electric moment of a system of 
charges. It is easy to verify that for a dipole Eq. (1.66) transforms 
into the expression p = q\ which we are already familiar with. 

If the total charge of a system is zero (2 <h = 0), the value of the 
dipole moment does not depend on our choice of the origin of coor- 


Electric Field in a Vacuum 


35 


dinates. To convince ourselves that this is true, let us take two 
arbitrary origins of coordinates O and O' (Fig. 1.17). The position 
vectors of the i-th charge conducted from these points are related 
as follows: 

r; = b + r* (1 *67) 

(what the vector b is is clear from the figure). With account taken 
of Eq. (1.67), the dipole moment in the system with the origin O' is 

p' = 2 = 2 <?t ( b + r i) = b 2 Vi + 2 7t' r ( 

The first addend equals zero (because 2 ~ 0). The second one 

is p — the dipole moment in a coordinate system with its origin at O. 
We have thus obtained that p' = p. 

Equation (1.65) is in essence the first two terms of the series expan- 
sion of function (1.63) by powers of r f /r. When 2 ?« ^ 0, the first 
term of Eq. (1.65) makes the main contribution to the potential 



Fig. 1.17 






Fig. 1.18 


(the second term diminishes in proportion to 1/r 2 and is therefore 
much smaller than the first one). For an electrically neutral system 
(2 = 0), the first term equals zero, and the potential is determined 

mainly by the second term of Eq. (1.65). This is how matters stand, 
in particular, for the field of a dipole. 

For the system of charges depicted in Fig. 1.18a and called 
a quadrupole, both 2 Qt and P equal zero so that Eq. (1.65) gives 
a zero value of the potential. Actually, however, the field of a quad- 
rupole, although it is much weaker than that of a dipole (with the 
same values of q and Z), differs from zero. The potential of the field 
set up by a quadrupole is determined mainly by the third term of 
the expansion that is proportional to 1/r 3 . To obtain this term, we 
must take into consideration quantities of the order of ( r t /r ) a which 
we disregarded in deriving Eq. (1.65). For the system of charges 
shown in Fig. 1.186 and called an octupole, the third term of the 
expansion also equals zero. The potential of the field of such a system 


36 


Electricity and Magnetism 


is determined by the fourth term of the expansion, which is propor- 
tional to 1/r 4 . 

It must be noted that the quantity equal to i n the numerator 
of the first term of Eq. (1.65) is called a monopole or a zero-order 
multipole, a dipole is also called a first-order multipole, a quadrupole 
is called a second-order multipole, and so on. 

Thus, in the general case, the field of a system of charges at great 
distances can be represented as the superposition of fields set up by 
multipoles of different orders — a monopole, dipole, quadrupole, 
octupole, etc. 


1.11. A Description of the Properties 
of Vector Fields 

To continue our study of the electric field, we must acquaint 
ourselves with the mathematical tools used to describe the proper- 
ties of vector fields. These tools are called vector analysis. In the 
present section, we shall treat the fundamental concepts and selected 
formulas of vector analysis, and also prove its two main theorems — 
the Ostrogradsky-Gauss theorem (sometimes called Gauss’s diver- 
gence theorem) and Stokes’s theorem. 

The quantities used in vector analysis can be best illustrated for 
the field of the velocity vector of a flowing liquid. We shall therefore 
introduce these quantities while dealing with the flow of an ideal 
incompressible liquid, and then extend the results obtained to 
vector fields of any nature. 

We are already acquainted with one of the concepts of vector 
analysis. This is the gradient, used to characterize scalar fields. 
If the value of the scalar quantity qp = <p (x, y, z) is compared 
with every point P having the coordinates x, y, z , we say that the 
scalar field of (p has been set. The gradient of the quantity <p is defined 
as the vector 

gr a d 9 = e* -f e „ + (1.68) 

The increment of the function <p upon displacement over the 
length dl — e x dx + dy + e z dz is 

d<p = -£-dx + - ^.dy + ^dz 

which can be written in the form 

dq> = V<p-dl (1.69) 

Now we shall go over to establishing the characteristics of vector 
fields. 



Electric Field In a Vacuum 


37 


Vector Flux. Assume that the flow of a liquid is characterized 
by the field of the velocity vector. The volume of liquid flowing 
in unit time through an imaginary surface S is called the flux of 
the liquid through this surface. To find the flux, let us divide the 
surface into elementary sections of the size AS. It can be seen from 
Fig. 1.19 that during the time At a volume of liq- 
uid equal to 

AV = (AS cos a) v At 

will pass through section AS. Dividing this volume 
by the time At, we shall find the flux through sur- 
face AS: 

A® = ASv cos a 

lit 

Passing over to differentials, we find that 

d<t> = (v cos a) dS (1-70) 

Equation (1.70) can be written in two other ways. First, if we take 
into account that v cos a gives the projection of the velocity vector 
onto the normal n to area dS, we can write Eq. (1.70) in the form 

dO = v n dS (1.71) 

Second, we can introduce the vector dS whose magnitude equals 
that of area dS, while its direction coincides with the direction 
of a normal n to the area: 



dS = dS»n 


Since the direction of the vector n is chosen arbitrarily (it can be 
directed to either side of the area), then dS is not a true vector, 
but is a pseudo vector. The angle a in Eq. (1.70) is the angle between 
the vectors v and dS. Hence, this equation can be written in the form 

dO = v dS (1.72) 


By summating the fluxes through all the elementary areas into 
which we have divided surface S, we get the flux of the liquid through 
S: 





(1.73) 


A similar expression written for an arbitrary vector field a, i.e. the 
quantity 

“ | a dS — ( d n dS 
b s 


(1.74) 


38 


Electricity and Magnetism 


is called the flux of the vector a through surface S. In accordance 
with this definition, the flux of a liquid can be called the flux of 
the vector v through the relevant surface [see Eq. (1 .73)). 

The flux of a vector is an algebraic quantity. Its sign depends 
on the choice of the direction of a normal to the elementary areas 
into which surface S is divided in calculating the flux. Reversal 
of the direction of the normal changes the sign of 
a n and, therefore, the sign of the quantity (1.74). 
The customary practice for closed surfaces is cal- 
culation of the flux “emerging outward” from the 
region enclosed by the surface. Accordingly, in 
the following we shall always implicate that n 
is an outward normal. 

We can give an illustrative geometrical in- 
terpretation of the vector flux. For this purpose, 
we shall represent a vector field by a system of 
lines a constructed so that the density of the lines 
at every point is numerically equal to the magnitude of the vector a 
at the same point of the field (compare with the rule for constructing 
the lines of the vector E set out at the end of Sec. 1.5). Let us find 
the number AN of intersections of the field lines with the imaginary 
area AS. A glance at Fig. 1.20 shows that this number equals the 
density of the lines (i.e. a) multiplied by AS ± = AS cos a: 

AN (=) a AS cos a = a n AS 

We are speaking only about the numerical equality between AN 
and a n AS. This is why the equality sign is confined in parentheses. 
According to Eq. (1.74), the expression a n AS is AO a — the flux 
of the vector a through area AS. Thus, 

AN (=) A<D a (1.75) 

For the sign of AN to coincide with that of AO tt , we must consider 
those intersections to be positive for which the angle a between the 
positive direction of a field line and a normal to the area is acute. 
The intersection should be considered negative if the angle a is 
obtuse. For the area shown in Fig. 1.20, all three intersections are 
positive: AN — +3 (AO a in this case is also positive because 
a n >0). If the direction of the normal in Fig. 1.20 is reversed, the 
intersections will become negative (AN = — 3), and the flux AO a 
will also be negative. 

Summation of Eq. (1.75) over the finite imaginary surface S 
yields 

<t> a (=) £ A7V = N+ - N, (1.76) 

where N+ and AL are the total number of positive and negative 
intersections of the field lines with surface S, respectively. 


AS 



Fig. 1.20 



Electric Field in a Vacuum 


39 


The reader may be puzzled by the circumstance that since the 
flux, as a rule, is expressed by a fractional number, the number of 
intersections of the field lines with a surface compared with the 
flux will also be fractional. Do not be confused by this, however. 
Field lines are a purely conditional image deprived of a physical 
meaning. 

Let us take an imaginary surface in the form of a strip of paper 
whose bottom part is twisted relative to the top one through the 



Fig. 1.21 



angle ji (Fig. 1.21). The direction of a normal must be chosen identi- 
cally for the entire surface. Hence, if in the top part of the strip 
a positive normal is directed to the right, then in the bottom part 
a normal will be directed to the left. Accordingly, the intersections 
of the field lines depicted in Fig. 1.21 with the top half of the surface 
must be considered positive, and with the bottom half, negative. 

An outward normal is considered to be positive for a closed surface 
(Fig. 1.22). Therefore, the intersections corresponding to outward 
protrusion of the lines (in this case the angle a is acute) must be 
taken with the plus sign, and the ones appearing when the lines 
enter the surface (in this case the angle a is obtuse) must be taken 
with the minus sign. 

Inspection of Fig. 1.22 shows that when the field lines enter a 
closed surface continuously, each line when intersecting the surface 
enters it and emerges from it the same number of times. As a result, 
the flux of the corresponding vector through this surface equals zero. 
It is easy to see that if field lines end inside a surface, the vector 
flux through the closed surface will numerically equal the difference 
between the number of lines beginning inside the surface (A be g) 
and the number f lines terminating inside the surface (A te rm) : 

' J C 

^o( = )^beg — Aterm (1.77) 


40 


Electricity and Magnetism, 


The sign of the flux depends on which of these numbers is greater. 
When ATfceg = ^tcrmi the flux equals zero. 

Divergence. Assume that we are given the field of the velocity 
vector of an incompressible continuous liquid. Let us take an imag- 
inary closed surface S in the vicinity of point 
P (Fig. 1.23). If in the volume confined by this 
surface no liquid appears and no liquid van- 
ishes, then the flux flowing outward through the 
surface will evidently equal zero. A liquid flux 
cD 0 other than zero will indicate that there are 
liquid sources or sinks inside the surface, i.e. 
points at which the liquid enters the volume 
(sources) or emerges from it (sinks). The mag- 
nitude of the flux determines the total alge- 
braic power of the sources and sinks*. When the sources predominate 
over the sinks, the magnitude of the flux will be positive, and when 
the sinks predominate, negative. 

The quotient obtained when dividing the flux <!>„ by the volume 
which it flows out from, i.e 



Fig. 1.23 


<t> v 

V 


(1.78) 


gives the average unit power of the sources confined in the vol- 
ume V . In the limit when V tends to zero, i.e. when the volume V 
contracts to point P, expression (1.78) gives the true unit power of 
the sources at point P, which is called the divergence of the vec- 
tor v (it is designated by div v). Thus, by definition, 

div v = lim-%- 
v-+p v 

The divergence of any vector a is determined in a similar way: 
div a = lim-^= lim &> adS (1-79) 

v-p v V-+P v * 

The integral is taken over arbitrary closed surface S surrounding 
point P**; V is the volume confined by this surface. Since the transi- 
tion V — ► P is being performed upon which S tends to zero, we can 
assume that Eq. (1.79) cannot depend on the shape of the surface. 
This assumption is confirmed by strict calculations. 

Let us surround point P with a spherical surface of an extremely 
small radius r (Fig. 124). Owing to the smallness of r, the volume V 


* The power of a source (sink) is defined as the volume of liquid discharged 
(absorbed) in unit time. A sink can be considered as a source with a negative 
power. 

** The circle on the integral sign signifies that integration is performed over 
a closed surface. 




Electric Field in a Vacuum 


41 


enclosed by the sphere will also be very small. We can therefore 
consider with a high degree of accuracy that the value of div a 
within the limits of the volume V is constant*. In this case, we 
can write in accordance with Eq. (1.79) that 

<D a « div a* V 

where O a is the flux of the vector a through the surface surrounding 
the volume V. By Eq. (1.77), O a equals the number of lines 



Fig. 1.24 Fig. 1.25 


of a beginning inside V if div a at point P is positive, or i^termt 
the number of lines of a terminating inside V if div a at point P 
is negative. 

It follows from the above that the lines of the vector a begin 
in the closest vicinity of a point with a positive divergence. The 
field .lines “diverge” from this point; the latter is the “source” of 
the field (Fig. 1.24a). On the other hand, in the vicinity of a point 
with a negative divergence, the lines of the vector a terminate. 
The field lines “converge” toward this point; the latter is the “sink” 
of the field (Fig. 1.246). The greater the absolute value of div a, 
the bigger is the number of lines that begin or terminate in the 
vicinity of the given point. 

It can be seen from definition (1.79) that the divergence is a scalar 
function of the coordinates determining the positions of points in 
space (briefly — a point function). Definition (1.79) is the most 
general one that is independent of the kind of coordinate system 
used. 

Let us find an expression for the divergence in a Cartesian coordi- 
nate system. We shall consider a small volume in the form of a paral- 
lelepiped with ribs parallel to the coordinate axes in the vicinity 
of point P (x y y , z) (Fig. 1.25). The vector flux through the surface 


• It is assumed that the value of div a changes continuously, without any 
jumps, when passing from one point of a field to another. 


42 


Electricity and Magnetism 


of the parallelepiped is formed from the fluxes passing through each 
of the six faces separately. 

Let us find the flux through the pair of faces perpendicular to the 
x-axis (in Fig. 1.25 these faces are designated by diagonal hatching 
and by the numbers 1 and 2). The outward normal n 2 to face 2 
coincides with the direction of the x-axis. Hence, for points of this 
face, a n% = a x . The outward normal n x to face 1 is directed opposite- 
ly to the x-axis. Therefore, for points on this face, a ni = — a x . 
The flux through face 2 can be written in the form 

a Xt2 AyAz 

where a x> 2 is the value of a x averaged over face 2 . The flux through 
face 1 is 

— a Xt t Ay Az 

where a x? x is the average value of a x for face 1 . The total flux through 
faces 1 and 2 is determined by the expression 

(a*. 2 — <*x. t) Ay Az (1.80) 

The difference a x% 2 — a x% i is the increment of the average (over 
a face) value of a x upon a displacement along the x-axis by Ax. 
Owing to the smallness of the parallelepiped (we remind our reader 
that we shall let its dimensions shrink to zero), this increment can 
be written in the form ( dajdx ) Ax, where the value dajdx is taken 
at point P*. Therefore, Ecr. (1.80) becomes ^ 

Similar reasoning allows us to obtain the following expressions for 
the fluxes through the pairs of faces perpendicular to the y- and 
z-axes: 

— -AF and AV 
dy dz 

Thus, the total flux through the entire close surface is determined 
by the expression 



Dividing this expression by AV y we shall find the divergence of the 
vector a at point P (x, y, z): 

* The inaccuracy which we tolerate here vanishes when the volume shrinks 
to point P in the limit transition. 




Electric Field in a Vacuum 


43 


The Ostrogradsky-Gauss Theorem. If we know the divergence of 
the vector a at every point of space, we can calculate the flux of 
this vector through any closed surface of finite dimensions. Let us 
first do this for the flux of the vector v (a liquid flux). The product 
of div v and dV gives the power of the sources of the liquid confined 

within the volume dV . The sum of such products, i.e. j div v*dF, 

gives the total algebraic power of the sources confined in the vol- 
ume V over which integration is performed. Owing to incompres- 
sibility of the liquid, the total power of the sources must equal the 
liquid flux emerging through surface S enclosing the volume V . 
We thus arrive at the equation 

<^> vdS= ^ divvdF 
s v 

A similar equation holds for a vector field of any nature: 

^>adS=jdivadF (1.82) 

S V 

This relation is called the Ostrogradsky-Gauss theorem. The integral 
in the left-hand side of the equation is calculated over an arbitrary 
closed surface S , and the integral in the right- 
hand side over the volume V enclosed by this 
surface. 

Circulation. Let us revert to the flow of an 
ideal incompressible liquid. Imagine a closed 
line — the contour I\ Assume that in some way 
or other we have instantaneously frozen the 
liquid in the entire volume except for a very 
thin closed channel of constant cross section 
including the contour T (Fig. 1.26). Depending 
on the nature of the velocity vector field, the liquid in the channel 
formed will either be stationary or move along the contour (circulate) 
in one of the two possible directions. Let us take the quantity equal 
to the product of the velocity of the liquid in the channel and the 
length of the contour^/ as a measure of this motion. This quantity is 
called the circulation of the vector v around the contour I\ Thus, 

circulation of v around T = ul 

(since we assumed that the channel has a constant cross section, 
the magnitude of the velocity v = const). 

At the moment when the walls freeze, the velocity component 
perpendicular to a wall will be eliminated in each of the liquid 
particles, and only the velocity component tangent to the contour 
will remain, i.e. u t . The momentum dpi is associated with this 
component. The magnitude of the momentum for a liquid particle 


i 0^1 fa ^ 




44 


Electricity and Magnetism 


contained within a segment of the channel of length dl is p av t dl 
(p is the density of the liquid, and a is the cross-sectional area of 
the channel). Since the liquid is ideal, the action of the walls can 
change only the direction of the vector dpi, but not its magnitude. 
The interaction between the liquid particles will cause a redistribu- 
tion of the momentum between them that will level out the veloci- 
ties of all the particles. The algebraic sum of the tangential com- 



Fig. 1.27 


Fig. 1.28 


ponents of the momenta cannot change: the momentum acquired 
by one of the interacting particles equals the momentum lost by 
the second particle. This signifies that 

p avl — ^ pori; z dl 

T 

where v is the circulation velocity, and v t is the tangential compo- 
nent of the liquid’s velocity in the volume a dl at the moment of 
time preceding the freezing of the channel walls. Cancelling pa, 
we get 

circulation of v around r = vl = ^ v% dl 

T 

The circulation of any vector a around an arbitrary closed contour T 
is determined in a similar way: 

circulation of a around T— § a dl = ^ a t dl (1.83) 

r r 

It may seem that for the circulation to be other than zero the 
vector lines must be closed or at least bent in some way or other 
in the direction of circumventing the contour. It is easy to see that 
this assumption is wrong. Let us consider the laminar flow of water 
in a river. The velocity of the water directly at the river bottom is 
zero and grows as we approach the surface of the water (Fig. 1.27). 
The streamlines (lines of the vector v) are straight. Notwithstanding 




Electric Field in a Vacuum 


45 


this fact, the circulation of the vector v around the contour depicted 
by the dash line obviously differs from zero. On the other hand, 
in a field with curved lines, the circulation may equal zero. 

Circulation has the property of additivity. This signifies that 
the sum of the circulations around contours and T 2 enclosing 
neighboring surfaces S x and So (Fig. 1.28) equals the circulation 
around contour T enclosing surface S , which is the sum of surfaces 
and S 2 . Indeed, the circulation C x around the 
contour bounding surface S x can be represented 
as the sum of the integrals 

2 1 

C, = §adl = J adl (1.84) 

T x 1 2 

(I) (lnt.) 

The first integral is taken over section I of the 
outer contour, the second over the interface be- 
tween surfaces S 1 and S 2 in direction 2-1. 

Similarly, the circulation C 2 around the contour enclosing sur- 
face S 2 is 

1 2 

'adl (1.85) 



C 2 = ^ a dl = j a dl + J 
r 2 2 i 


(ID 


(int.) 


The first integral is taken over section II of the outer contour, the 
second over the interface between surfaces S x and S 2 in direction 1-2 . 

The circulation around the contour bounding total surface S can 
be represented in the form 


C = 


<^> adl 
r 


2 1 

= j a dl -f- j a dl 

1 2 

(i) (ii) 


( 1 . 86 ) 


The second addends in Eqs. (1.84) and (1.85) differ only in their 
sign. Therefore, the sum of these expressions will equal Eq. (1.86). 
Thus, 

C - C x + C 2 (1.87) 


Equation (1.87) which we have proved does not depend on the 
shape of the surfaces and holds for any number of addends. Hence, 
if we divide an arbitrary open surface S into a great number of 
elementary surfaces AS* (Fig. 1.29), then the circulation around 
the contour enclosing S can be written as the sum of the elementary 


* In the figure, the elementary surfaces are depicted in the form of rectangles. 
Actually, their shape may be absolutely arbitrary 



46 


Electricity and Magnetism 


circulations AC around the contours enclosing the AS's: 

c = S AC, (1.88) 

Curl. The additivity of the circulation permits us to introduce 
the concept of unit circulation, i.e. consider the ratio of the circula- 
tion C to the magnitude of surface S around which the circulation 
“flows”. When surface S is finite, the ratio C/S gives the mean value 
of the unit circulation. This value characterizes the properties of 
a field averaged over surface S . To obtain the characteristic of the 
field at point P, we must reduce the dimensions of the surface, 
making it shrink to point P . The ratio C/S tends to a limit that 
characterizes the properties of the field at point P. 

Thus, let us take an imaginary contour T in a plane passing through 
point P, and consider the expression 

lim % (1.89) 

S-+P 

where C a = circulation of the vector a around the contour T 
S = surface area enclosed by the contour. 

Limit (1.89) calculated for an arbitrarily oriented plane cannot 
be an exhaustive characteristic of the field at point P because the 
magnitude of this limit depends on the orientation of the contour 
in space in addition to the properties of the field at point P. This 
orientation can be given by the direction of a positive normal n 
to the plane of the contour (a positive normal is one that is associat- 
ed with the direction of circumvention of the contour in integration 
by the right-hand screw rule). In determining limit (1.89) at the 
same point P for different directions n, we shall obtain different 
values. For opposite directions, these values will differ only in their 
sign (reversal of the direction n is equivalent to reversing the direc- 
tion of circumvention of the contour in integration, which only 
causes a change in the sign of the circulation). For a certain direction 
of the normal, the magnitude of expression (1.89) at the given point 
will be maximum. 

Thus, quantity (1.89) behaves like the projection of a vector onto 
the direction of a normal to the plane of the contour around which 
the circulation is taken. The maximum value of quantity (1.89) 
determines the magnitude of this vector, and the direction of the 
positive normal n at which the maximum is reached gives the direc- 
tion of the vector. This vector is called the curl of the vector a. 
Its symbol is curl a. Using this notation, we can write expression 
(1.89) in the form 

(curl a) n = lim -%- = lim (f) a dl 
S-+P b s—p b 


(1.90) 


Electric Field in a Vacuum 


47 


We can obtain a graphical picture of the curl of the vector v by 
imagining a small and light fan impeller placed at the given point 
of a flowing liquid (Fig. 1.30). At the spots where the curl differs 
from zero, the impeller will rotate, its velocity being the higher, 
the greater in value is the projection of the curl onto the impeller 
axis. 

Equation (1.90) defines the vector curl a. This definition is a most 
general one that does not depend on the kind of coordinate system 
used. To find expressions for the projections of the vector curl a 
onto the axes of a Cartesian coordinate system, we must determine 
the values of quantity (1.90) for such orientations of area S for 




Fig. 1.30 


Fig. 1.31 


which the normal n to the area coincides with one of the axes x, y, z. 
If, for example, we direct n along the x-axis, then (1.90) becomes 
(curl a) x . Contour P in this case is arranged in a plane parallel 
to the coordinate plane yz. Let us take this contour in the form of 
a rectangle with the sides. Ay and A z (Fig. 1.31, the x-axis is direct- 
ed toward us in this figure; the direction of circumvention indicated 
in the figure is associated with the direction of the x-axis by the 
right-hand screw rule). Section 1 of the contour is opposite in direc- 
tion to the z-axis. Therefore, a t on this section coincides with — a z . 
Similar reasoning shows that a t on sections 2 , 5 , and 4 equals a yy a Zy 
and — a yy respectively. Hence, the circulation can be written in 
the form 

(a Zf 3 a 2, i) Az ( o.y , 4 dy t 2 ) Ay (1.91) 

where a z 3 and a z l are the average values of a z on sections 3 and l f 
respectively, and a y%k and a y % are the average values of a y on 
sections 4 and 2. 

The difference a Zt9 — a z l is the increment of the average value 
of a z on the section Az when this section is displaced in the direction 
of the y-axis by Ay. Owing to the smallness of Ay and Az, this incre- 
ment can be represented in the form ( dajdy ) Ay, where the value of 



48 


Electricity and Magnetism 


dajdy is taken for point P*. Similarly, the difference a y 4 — a y 2 
can be represented in the form ( da y !dz ) A z. Using these expressions 
in Eq. (1.91) and putting the common factor outside the parentheses, 
we get the following expression for the circulation: 


/ da L 
\ ~dy 




da z 

dy 



where A S is the area of the contour. Dividing the circulation by 
AS , we find the expression for the projection of curl a onto the 
x-axis: 


(curl a) t = 


We can 


(curl a) z 


1 

o 

N 

da y 

dy 

dz 

ing that 


_ da x 

da z 

dz 

dx 

_ 

da x 

: dx 

dy 


(1.92) 

(1.93) 

(1.94) 


It is easy to see that any of the equations (1.92)-(1.94) can be 
obtained from the preceding one [Eq. (1.94) should be considered 
as the preceding one for Eq. (1.92)1 by the so-called cyclic transpo- 
sition of the coordinates, i.e. by replacing the coordinates according 
to the scheme 


U 


Thus, the curl of the vector a is determined in the Cartesian coor- 
dinate system by the following expression: 


curl a = e x ( 


da T 


da u 


dy 


dz 


)+M 


da x 


dz 


da z 

dx 


)+e* ( 


da n 


day 


dx 


dy 


) (1.95) 


Below we shall indicate a more elegant way of writing this expres- 
sion. 

Stokes’s Theorem. Knowing the curl of the vector a at every point 
of surface S (not necessarily plane), we can calculate the circulation 
of this vector around contour T enclosing S (the contour may also 
not be plane). For this purpose, we divide the surface into very small 
elements AS. Owing to their smallness, these elements can be con- 
sidered as plane. Therefore in accordance with Eq. (1.90), the cir- 


* The inaccuracy which we tolerate here vanishes when the contour shrinks 
to point P in the limit transition. 



Electric Field in a Vacuum 


49 


culation of the vector a around the contour bounding AS can be 
written in the form 

A C « (curl a) n AS = curl a* AS (1.96) 

where n is a positive normal to surface element AS. 

In accordance with Eq. (1.88), summation of expression (1.96) 
over all the A S’s yields the circulation of the vector a around con- 
tour T enclosing S: 

C = 2 AC ^ 2 cur l AS 

Performing a limit transition in which all the A S’s shrink to zero 
(their number grows unlimitedly), we arrive at the equation 

a dl = j curl a • dS (1 .97) 

r s 


§ 


Equation (1.97) is called Stokes’s theorem. Its meaning is that 
the circulation of the vector a around an arbitrary contour V equals 
the flux of the vector curl a through the arbitrary surface S surrounded 
by the given contour . 

The Del Operator. Writing of the formulas of vector analysis is 
simplified quite considerably if we introduce a vector differential 
operator designated by the symbol V (nabla or del) and called the 
del operator or. the Hamiltonian operator. This operator denotes 
a vector with the components dfdx , dldy, and d/dz. Consequently, 

V = e,^ + e,-| r + e,|- (!.98) 


This vector has no meaning by itself. It acquires a meaning in 
combination with the scalar or vector function by which it is sym- 
bolically multiplied. Thus, if we multiply the vector V by the 
scalar <p, we obtain the vector 


V<P ~ Cj; 


a<£_ iSL 

dx dy 



(1.99) 


which is the gradient of the function cp (see Eq. (1.68)], 

The scalar product of the vectors V and a gives the scalar 


Va = Vx a x + Vyfly + V z a z = 


da x 

dx 


da u da, 
~df"' >r ~dz 


1 . 100 ) 


which we can see to be the divergence of the vector a (see Eq. (1.81)1- 
Finally, the vector_product of the vectors V and a gives a vector 
with the components (Val* = VyO* — V z 2 „ — dajdy — da y /dz, 
etc., that coincide with the components of curl a (see Eqs. (1.92)- 
(1.94)1. Hence, using the writing of a vector product with the aid 



50 


Electricity and Magnetism 


of a determinant, we have 


curl a = [Va] = 


d 

dx 


d 

dy 

a., 


d 

dz 

a z 


( 1 . 101 ) 


Thus, there are two ways of denoting the gradient, divergence, 
and curl: 


V<p = gradq>, Va = diva, [Va]s=curla 


The use of the del symbol has a number of advantages. We shall 
therefore use such symbols in the following. One must accustom 
oneself to identify the symbol V9 with the words “gradient of phi” 
(i.e. to say not “del phi”, but “gradient of phi”), the symbol Va 
with the words “divergence of a” and, finally, the symbol [val 
with the words “curl of a”. 

When using the vector V, one must remember that it is a dif- 
ferential operator acting on all the functions to the right of it. 
(Consequently, in transforming expressions including V, one must 
'take into consideration both the rules of vector ^algebra and those 
of differential calculus. For example, the derivative of the product 
of the functions <p and 9 is 

MO' = + tp'l’' 

Accordingly, 


grad (99) = V (<p^) = tV<P + 9V9 = 

= 9 grad <p + <p grad 9 (1.102) 

Similarly 

div (<pa) = V (9a) = aV<p + 9Va = a grad <p + 9 div a (1.103) 


The gradient of a function <p is a vector function. Therefore, the 
divergence and curl operations can be performed with it: 


div grad q> = V ( V<p) « ( V V) 9 = ( Vi + VJ + Vj) 9 = 

--&+-S-+-&- 4 * < iio4 > 

(A is the Laplacian operator) 

curl grad 9 = [v, V9I = fwl 9 — 0 ( 1 . 105 ) 

(we remind our reader that the vector product of a vector and itself 
is zero). 

Let us apply the divergence and curl operations to the function 
curl a: 

div curl a = V IVal = 0 ( 1 . 106 ) 



Electric Field in a Vacuum 


51 


(a scalar triple product equals the volume oi a parallelepiped con- 
structed on the vectors being multiplied (see Vol. I, p. 31); if 
two of these vectors coincide, the volume of the parallelepiped equals 
zero); 

curl curl a = lv, tVall = V (Va) — (W) a = 

= grad div a — Aa (1.107) 

[we have used Eq. (1.35) of Vol. I, p. 32, namely, [a, [bell * 
= b (ac ) — c (ab)J. 

Equation (1.106) signifies that the field of a curl has no sources. 
Hence, the lines of the vector lVal have neither a beginning nor 
an end. It is exactly for this reason that the flux of a curl through 
any surface S resting on the given contour T is the same [see Eq. (1.97)]. 

We shall note in concluding that when the del operator is used, 
Eqs. (1.82) and (1.97) can be given the form 

a • dS = j Va -dV (the Ostrogradsky-Gauss theorem) (1.408) 
^ a -dl = | [Va] -dS (Stokes's theorem) (1.109) 


1.12. Circulation and Curl 

of an Electrostatic Field 

We established in Sec. 1.6 that the forces acting on the charge q 
in an electrostatic field are conservative. Hence, the work of these 
forces on any closed path T is zero: 

A = ^ gE dl = 0 
r 

Cancelling q, we get 

§Edl = 0 (1.110) 

r 

(compare with Eq. (1.46)]. 

The integral in the left-hand side of Eq. (1.110) is the circulation 
of the vector E around contour T [see expression (1.80)]. Thus, an 
electrostatic field is characterized by the fact that the circulation 
of the strength vector of this field around any closed contour equals 
zero. 

Let us take an arbitrary surface S resting on contour T for which 
the circulation is calculated (Fig. 1.32). According to Stokes’s 
theorem [see Eq. (1.109)], the integral of curl E taken over this 


52 


Electricity and Magnetism 


surface equals the circulation of the vector E around contour T: 



( 1 . 111 ) 


Since the circulation equals zero, we arrive at the conclusion that 

| (VE]dS = 0 


This condition must be observed for any surface S resting on arbitrary 
contour I\ This is possible only if the curl of the vector E at 
every point of the field equals zero: 

IVE) = 0 (1.112) 

By analogy with the fan impeller shown in Fig. 1.25, let us ima- 
gine an electrical “impeller” in the form of a light hub with spokes 



E 

Fig. 1.32 Fig. 1.33 



whose ends carry identical positive charges q (Fig. 1.33; the entire 
arrangement must be small in size). At the points of an electric 
field where curl E differs from zero, such an impeller would rotate 
with an acceleration that is the greater, the larger is the projection 
of the curl onto the impeller axis. For an electrostatic field, such 
an imaginary arrangement would not rotate with any orientation 
of its axis. 

Thus, a feature of an electrostatic field is that it is a non-circuital 
one. We established in the preceding section that the curl of the 
gradient of a scalar function equals zero (see expression (1.96)]. 
Therefore, the equality to zero of curl E at every point of a field 
makes it possible to represent E in the form of the gradient of a sca- 
lar function <p called the potential. We have already considered this 
representation in Sec. 1.8 (see Eq. (1.41); the minus sign in this 
equation was taken from physical considerations]. 

We can immediately conclude from the need to observe condi- 
tion (1.110) that the existence of an electrostatic field of the kind 
shown in Fig. 1.34 is impossible. Indeed, for such a field, the circu- 



Electric Field in a Vacuum 


53 


lation around the contour shown by the dash line would differ 
from zero, which contradicts condition (1.110). It is also impossible 
for a field differing from zero in a restricted volume to be homogeneous 




E 


Fig. 1.34 


Fig. 1.35 


throughout this volume (Fig. 1.35). In this case, the circulation 
around the contour shown by the dash line woiild differ from zero. 


<t)f # ^ 




V A ur ^ 
Wllfo > £ V i * 


1.13. Gauss’s Theorem 


rt 


r ^7 

A - 


, * 

$ 

' 'IV 


We established in the preceding section what the curl of an elec- 
trostatic field equals. Now let us find the divergence of a field. For 
this purpose, we shall consider the field of 
a point charge q and calculate the flux of 
the vector E through closed surface S sur- 
rounding the charge (Fig. 1.36). We showed 
in Sec. 1.5 that the number of lines of the 
vector E beginning at a point charge -\-q or 
terminating at a charge — q numerically 
equals q/e 0 . 

By Eq. (1.77), the flux of the vector E 
through any closed surface equals the num- 
ber of lines coming out, i.e. beginning on the 
charge, if it is positive, and the number of 
lines entering the surface, i.e. terminating 
on the charge, if it is negative. Taking into 

account that the number of lines beginning or terminating at a point 
charge numerically equals g/e 0 (see Sec. 1.5), we can write that 



Fig. 1.36 


The sign of the flux coincides with that of the charge q. The dimen- 
sions of both sides of Eq. (1.113) are identical. 

Now let us assume that a closed surface surrounds N point charges 
q u q * t - • .. Qn- On the basis of the superposition principle, the 
strength E of the field set up by all the charges equals the sum of the 


54 


Electricity and Magnetism 


strengths E t set up by each charge separately: E = 2 E|- Hence, 

<D*=$EdS = ^ (2 E,)dS = 2 § E i<® 

S Si i s 

Each of the integrals inside the sum sign equals qj c 0 . Therefore 

N 

tD, = ^EdS = ^-2 ?« (1.114) 

8 * i=l 

The statement we have proved is called Gauss’s theorem. Accord- 
ing to it, the flux of an electric field strength vector through a closed 
surface equals the algebraic sum of the charges enclosed by this surface 
divided by e 0 . 

When considering fields set up by macroscopic charges (i.e. 
charges formed by an enormous number of elementary charges), 
the discrete structure of these charges is disregarded, and they are 
considered to be distributed in space continuously with a finite 
density everywhere. The volume density of a charge p is determined 
by analogy with the density of a mass as the ratio of the charge dq 
to the infinitely small (physically) volume dV containing this 
charge: 

P = -^£- (1.115) 

In the given case by an infinitely small (physically) volume, we 
must understand a volume which on the one hand is sufficiently 
small for the density within its limits to be considered identical, 
and on the other is sufficiently great for the discreteness of the 
charge not to manifest itself. 

Knowing the charge density at every point of space, we can find 
the total charge surrounded by closed surface S. For this purpose, 
we must calculate the integral of p with respect to the volume enclosed 
by the surface: 

2?i = jp<*F 

Thus, Eq. (1.114) can be written in the form 

§EdS = -l-jpdF (1.116) 

Replacing the surface integral with a volume one in accordance 
with Eq. (1.108), we have 

jvEdV = -LjpdV 



Electric Field in a Vacuum 


55 


The relation which we have arrived at must be observed for any 
arbitrarily chosen volume V . This is possible only if the values of 
the integrands for every point of space are the same. Hence, the 
divergence of the vector E is associated with the density of the 
charge at the same point by the equation 

VE=4- p (1.117) 

c o 

This equation expresses Gauss’s theorem in the differential form. 

For a flowing liquid, Vv gives the unit power of the sources of the 
liquid at a given point. By analogy, charges are said to be sources 
of an electric field. 


1.14. Calculating Fields with the Aid of 
Gauss's Theorem 

Gauss’s theorem permits us in a number of cases to find the strength 
of a field in a much simpler way than by using Eq. (1.15) for the 
field strength of a point charge and the field superposition principle. 
We shall demonstrate the possibilities of Gauss’s theorem by employ- 
ing a few examples that will be useful for our 'further exposition. 
Before starting on our way, we shall introduce the concepts of 
surface and linear charge densities. 

If a charge is concentrated in a thin surface layer of the body 
carrying the charge, the distribution of the charge in space can be 
characterized by the surface density a, which is determined by the 
expression 



Here dq is the charge contained in the layer of area dS . By dS is 
meant an infinitely small (physically) section of the surface. 

If a charge is distributed over the volume or surface of a cylindri- 
cal body (uniformly in each section), the linear charge density is 
used, i.e. 


= (1.H9) 

where dl — length of an infinitely small (physically) segment of 
the cylinder 

dq = charge concentrated on this segment. 

Field of an Infinite Homogeneously Charged Plane. Assume that 
the surface charge density at all points of a plane is identical and 



56 


Electricity and Magnetism 


equal to a; for definiteness we shall consider the charge to be positive. 
It follows from considerations of symmetry that the field strength 
at any point is directed at right angles to the plane. Indeed, since 
the plane is infinite and charged homogeneously, there is no reason 
why the vector E should deflect to a side from a normal to the plane. 
It is further evident that at points symmetrical relative to the 
plane, the field strength is identical in magnitude and opposite in 
direction. 

Let us imagine mentally a cylindrical surface with generatrices 
perpendicular to the plane and bases of a size A S arranged symmetri- 



Fig. 1.37 


Fig. 1.38 


cally relative to the plane (Fig. 1.37). Owing to symmetry, we have 
E' = E” — E. We shall apply Gauss’s theorem to the surface. 
The flux through the side part of the surface will be absent because 
E n at each point of it is zero. For the bases, E n coincides with E . 
Hence, the total flux through the surface is 22? AS. The surface 
encloses the charge crAS. According to Gauss’s theorem, the condi- 
tion must be observed that 


2EAS = 


®0 


whence 



( 1 . 120 ) 


The result we have obtained does not depend on the length of the 
cylinder. This signifies that at any distances from the plane, the 
field strength is identical in magnitude. The field lines are shown 
in Fig. 1.38. For a negatively charged plane, the result will be the 
same except for the reversal of the direction of the vector E and 
the field lines. 




Electric Field in a Vacuum 


57 


If we take a plane of finite dimensions, for instance a charged 
thin plate*, then the result obtained above will hold only for points, 
the distance to which from the edge of the plate considerably exceeds 
the distance from the plate itself (in Fig. 1.39 the region containing 
such points is outlined by a dash line). At points at an increasing 
distance from the plane or approaching its edges, the field will differ 
more and more from that of an infinitely charged plane. It is easy 



Fig. 1.39 


+ & -o' 



Fig. 1.40 


to imagine the nature of the field at great distances if we take into 
account that at distances considerably exceeding the dimensions 
of the plate, the field it sets up can be treated as that of a point 
charge. 

Field of Two Uniformly Charged Planes. The field of two parallel 
infinite planes carrying opposite charges with a constant surface 
density or identical in magnitude can be found by superposition 
of the fields produced by each plane separately (Fig. 1.40). In the 
region between the planes, the fields being added have the same 
direction, so that the resultant field strength is 

(M21) 

Outsid e the volume bounded by the planes, the fields being added 
have opposite directions so that the resultant field strength equals 
zero. 

Thus, the field is concentrated between the planes. The field 
strength at all points of this region is ident ical in value and in direc- 


* For a plate, by o in Eq. (1.120) should be understood the charge concentrat- 
ed on 1 m 3 of the plate over its entire thickness. In metal bodies, the charge is 
distributed over the external surface. Therefore by a we should understand the 
double value of the charge density on the surfaces surrounding the metal plate. 


58 


Electricity and Magnetism 


tion; consequently, the field is homogeneous. The field lines are 
a collection of parallel equispaced straight lines. 

The result we have obtained also holds approximately for planes 
of finite dimensions if the distance between them is much smaller 
than their linear dimensions (a parallel-plate capa- 
citor). In this case, appreciable deviations of the 
field from homogeneity axe observed only near the 
edges of the plates (Fig. 1.41). 

Field of an Infinite Charged Cylinder. Assume 
that the field is produced by an infinite cylin- 
drical surface of radius R whose charge has a con- 
stant surface density a. Considerations of symmetry 
show that the field strength at any point must be 
directed along a radial line perpendicular to the cyl- 
inder axis, and that the magnitude of the strength 
can depend only on the distance r from the cylin- 
der axis. Let us mentally imagine a coaxial closed 
cylindrical surface of radius r and height h with a 
charged surface (Fig. 1.42). For the bases of the cyl- 
inder, we have E n = 0, for the side surface E n = 
Fig. 1.41 =E (r) (the charge is assumed to be positive).' 

Hence, the flux of the vector E through the surface 
being considered is E (r)-2nrA. If r >R, the charge q = Xh (where 
K is the linear charge density) will get into the surface. Applying 
Gauss’s theorem, we find that 

E(r)- 2nrA = — 

Co 

Hence, 

<'•>*> <*- i22 > 

If r < R, the closed surface being 
considered contains no charges inside, 
owing to which E ( r ) = 0. 

Thus, there is no field inside a uniformly charged cylindrical 
surface of infinite length. The field strength outside the surface is 
determined by the linear charge density X and the distance r from 
the cylinder axis. 

The field of a negatively charged cylinder differs from that of 
a positively charged one only in the direction of the vector E. 

A glance at Eq. (1.122) shows that by reducing the cylinder 
radius R (with a constant linear charge density X), we can obtain 
a field with a very great strength near the surface of the cylinder. 

Introducing X = 2 nRo into Eq. (1.122) and assuming that r = 
= R, we 'get the following value for the field strength in direct 






Electric Field in a Vacuum 


50 


proximity to the surface of a cylinder: 

E(R) = ± (1.123) 

The superposition principle makes it simple to find the field of 
two coaxial cylindrical surfaces carrying a linear charge density > 
of the same magnitude, but of opposite signs (Fig. 1.43). .There is 
no field inside the smaller and outside the larger cylinders. The 
field strength in the gap between the 
cylinders is determined by Eq. (1.122). 

This also holds for cylindrical surfaces 
of a finite length if the gap between the 
surfaces is much smaller than their 
length (a cylindrical capacitor). Appre- 
ciable deviations from the field of sur- 
faces of an infinite length will be ob- 
served only near the edges of the 
cylinders. 

Field of a Charged Spherical Surface. 

The field produced by a spherical sur- 
face of radius R whose charge has a 
constant surface density a will obvi- 
ously be a centrally symmetrical one. 

This signifies that the direction of the vector E at any point passes 
through the centre of the sphere, while the magnitude of the field 
strength is a function of the distance r from the centre of the sphere. 
Let us imagine a surface of radius r that is concentric with the charged 
sphere. For all points of this surface, E n = E (r). If r the 

entire charge q distributed over the sphere will be inside the sur- 
face. Hence, 

E(r) 4*r*=-£- 
e 0 

whence 

= ( r > R ) t 1 - 124 ) 

A spherical surface of radius r less than R will contain no charges, 
owing to which for r < R we get E (r) = 0. 

Thus, there is no field inside a spherical surface whose charge 
has a constant surface density o . Outside this surface, the field 
is identical with that of a point charge of the same magnitude at the 
centre of the sphere. 

Using the superposition principle, it is easy to show that the 
field of two concentric spherical surfaces (a spherical capacitor) 
carrying charges +<7 and — q that are identical in magnitude and 
opposite in sign is concentrated in the gap between the surfaces. 



Fig. 1.43 



60 


Electricity and Magnetism 


the magnitude of the field strength in the gap being determined by 
Eq. (1.124). 

Field of a Volume-Charged Sphere. Assume that a sphere of radius 
R has a charge with a constant volume density p. The field in this 
case has central symmetry. It is easy to see that the same result 
is obtained for the field outside the sphere [see Eq. (1.124)] as for 
a sphere with a surface charge. The result will be different for points 
inside the sphere, however. A spherical surface of radius r (r < R) 

contains a charge equal to p* —-Jir 3 . Therefore, Gauss’s theorem for 

u 

such a surface will be written as follows: 

E(r) 4nr* = J-pA nr » 

Hence, substituting qj for p, we get 

BV-tk-k' <-•<*> < 1126 > 

Thus, the field strength inside a sphere grows linearly with the 
distance r from the centre of the sphere. Outside the sphere, the 
field strength diminishes according to the same law as for the field 
of a point charge. 




CHAPTER 2 ELECTRIC FIELD 

IN DIELECTRICS 


2.1. Polar and Non-Polar Molecules 

Dielectrics (or insulators) are defined as substances not capable 
of conducting an electric current. Ideal insulators do not exist in 
nature. All substances, even if to a negligible extent, conduct an 
electric current. But substances called conductors conduct a cur- 
rent from 10“ to 10 ao times better than substances called dielectrics. 

If a dielectric is introduced into an electric field, then the field 
and the dielectric itself undergo appreciable changes. To understand 
why this happens, we must take into account that atoms and mole- 
cules contain positively charged nuclei and negatively charged 
electrons. 

A molecule is a system with a total charge of zero. The linear 
dimensions of this system are very small, of the order of a few ang- 
stroms (the angstrom — A — is a unit of length equal to 10” 10 m that 
is very convenient in atomic physics). We established in Sec. 1.10 
that the field set up by such a system is determined by the magnitude 
and orientation of the dipole electric moment 

P = 2 (21) 

(summation is performed both over the electrons and over the 
nuclei). True, the electrons in a molecule are in motion, so that 
this moment constantly changes. The velocities of the electrons 
are so high, however, that the mean value of the moment (2.1) is 
detected in practice. For this reason in the following by the dipole 
moment of a molecule, we shall mean the quantity 

P = <r«> (2 2) 

(for nuclei, r* is simply taken as (r f -> in this sum). In other words, 
we shall consider that the electrons are at rest relative to the nuclei 
at certain points obtained by averaging the positions of the electrons 
in time. 

The behaviour of a molecule in an external electric field is also 
determined by its dipole moment. We can verify this by calculating 
the potential energy of a molecule in an external electric field. 
Selecting the origin of coordinates inside the molecule and taking 
advantage of the smallness of (rj), let us write the potential at the 
point where the i-th charge is in the form 

<Pi = <P + V«p- <r f > 



62 


Electricity and Magnetism 


where <p is the potential at the origin of coordinates [see Eq. (1.69)]* 
Hence, 

Wp— 2 q,q>i = 23 ?t (<P + Vq>- <n>) = <p2ft+V<P 2 <ri> 

Taking into account that 2 Qi = 0 and substituting — E for V<J>, 
we get 

W p = — E (ri> = — pE = —pE cos a 

Differentiating this expression with respect to a, we get Eq. (1.57) 
for the rotational moment; differentiating with respect to :r, we 
arrive at the force (1.62). 

Thus, a molecule is equivalent to a dipole both with respect to 
the field it sets up and with respect to the forces it experiences 
in an external field. The positive charge of this dipole equals the 
total charge of the nuclei and is at the “centre of gravity” of the 
positive charges; the negative charge equals the total charge of the 
electrons and is at the “centre of gravity” of the negative charges. 

In symmetrical molecules (such as H 2 , 0 2 , N t ), the centres of 
gravity of the positive and negative charges coincide in the absence 
of an external electric field. Such molecules have no intrinsic dipole 
moment and are called non-polar. In asymmetrical molecules (such 
as CO, NH, HC1), the centres of gravity of the charges, of opposite 
signs are displaced relative to each other. In this case, the mole- 
cules have an intrinsic dipole moment and are called polar. 

Under the action of an external electric field, the charges in 
a non-polar molecule become displaced relative to one another, 
the positive ones in the direction of the field, the negative ones 
against the field. As a result, the molecule acquires a dipole moment 
whose magnitude, as shown by experiments, is proportional to the 
field s trength. In the rationalized system, the constant of proportion- 
ality is written in the form e<>p, where e 0 is the electric constant, 
and p is a quantity called the polarizability of a molecule. Since 
the directions of p and E coincide, we can write that 

p = pe 0 E (2.3) 

The dipole moment has a dimension of [ 9 ] L. By Eq. (1.15), the 
dimension of e^E is [q] L“ 2 . Hence, the polarizability of a molecule p 
has the dimension L s . 

The process of polarization of a non-polar molecule proceeds as 
if the positive and negative charges of the molecule were bound to 
one another by elastic forces. A non-polar molecule is therefore said 
to behave in an external field like an elastic dipole. 

The action of an external field on a polar molecule consists mainly 
in tending to rotate the molecule so that its dipole moment is ar- 
ranged in the direction of the field. An external field does not virtu- 
ally affect the magnitude of a dipole moment. Consequently, a 
polar molecule behaves in an external field like a rigid dipole. 


Electric Field in Dielectrics 


63 


2.2. Polarization of Dielectrics 


In the absence of an external electric field, the dipole moments 
of the molecules of a dielectric usually either equal zero (non-polar 
molecules) or are distributed in space by directions chaotically 
(polar molecules). In both cases, the total dipole moment of a dielec- 
tric equals zero*. 

A dielectric becomes polarized under the action of an external 
field. This signifies that the resultant dipole moment of the dielectric 
becomes other than zero. It is quite natural to take the dipole mo- 
ment of a unit volume as the quantity characterizing the degree of 
polarization. If the field or the dielectric (or both) are not homoge- 
neous, the degrees of polarization at different points of the dielectric 
will differ. To characterize the polarization at a given point, we 
must separate an infinitely small (physically) volume AF containing 

this point, find the sum 2 P °f the moments of the molecules 

av 

confined in this volume, and take the ratio 



(2.4) 


The vector quantity P defined by Eq. (2.4) is called the polarization 
of a dielectric. 

The dipole moment p has the dimension [g] L. Consequently, 
th6 dimension of P is (^J L“ 2 , i.e. it coincides with the dimension 
of e 0 E [see Eq. (1.15)1. 

The polarization of isotropic dielectrics of any kind is associated 
with the field strength at the same point by the simple relation 

P = X*oE (2.5) 

where X is a quantity independent of E called the electric suscep- 
tibility of a dielectric**. It was indicated above that the dimensions 
of P and e 0 E are identical. Hence, X is a dimensionless quantity. 


• In Sec. 2.9, we shall acquaint ourselves with substances that can have a 
dipole moment in the absence of an external field. 

** In anisotropic dielectrics, the directions of P and E, generally speaking, 
do not coincide. In this case, the relation between P and E is described by the 
equations 

^ x“6o (Xxx& x-\~XxyEy -b XxtEj) 

P |/ = £ 0 (Xj/x^x + % yyE y~\~XvzE z) 

Pz = e o (XzxEx + XryEy + XizEz) 

The combination of the nine quantities Xih fo rms a symmetrical tensor of rank 
two called the tensor of the dielectric susceptibility [compare with Eqs. (5.30) 
of Vol. I, p. 147). This tensor characterizes the electrical properties of an aniso- 
tropic dielectric. 


64 


Electricity and Magnetism 


In the Gaussian system of units, Eq. (2.5) has the form 

P = (2.6) 

For dielectrics built of non-polar molecules, Eq. (2.5) issues from 
the following simple considerations. The volume AV contains 
a number of molecules equal to n AV, where n is the number of 
molecules per unit volume. Each of the moments p is determined 
in this case by Eq. (2.3). Hence, 

2 p = nAFpe 0 E 

AV 

Dividing this expression by AV, we get the polarization P = 
= npe 0 E. Finally, introducing the symbol ^ = n(5, we arrive at 
Eq. (2.5). 

For dielectrics built of polar molecules, the orienting action of the 
external field is counteracted by the thermal motion of the mole- 
cules tending to scatter their dipole moments in all directions. As 
a result, a certain preferred orientation of the dipole moments of 
the molecules sets in in the direction of the field. The relevant 
statistical calculations, which agree with experimental data, show 
that the polarization is proportional to the field strength, i.e. leads 
to Eq. (2.5). The electric susceptibility of such dielectrics varies 
inversely with the absolute temperature. 

In ionic crystals, the separate molecules lose their individuality. 
An entire crystal is, as it were, a single giant molecule. The lattice 
of an ionic crystal can be considered as two lattices inserted into 
each other, one of which is formed by the positive, and the other 
by the negative ions. When an external field acts on the crystal 
ions, both lattices are displaced relative to each other, which leads 
to polarization of the dielectric. The polarization in this case too is 
associated with the field strength by Eq. (2.5). We must note that 
the linear relation between E and P described by Eq. (2.5) may be 
applied only to not too strong fields [a similar remark relates to 
Eq. (2.3)]. 


2.3. The Field Inside a Dielectric 

The charges in the molecules of a dielectric are called bound. 
The action of a field can only cause bound charges to be displaced 
slightly from their equilibrium positions; they cannot leave the 
molecule containing them. 

Following the example of L. Landau and E. Lifshitz*, we shall 
call charges that, although they are within the boundaries of a di* 


* See L. D. Landau and E. M. Lifshitz. Elektrodinamika sploshnykh sred 
(Electrodynamics of Continuous Media). Moscow, Gostekhizdat (1957), p. 57. 




Electric Field in Dielectrics 


65 


electric, are not inside its molecules, and also charges outside a di- 
electric, extraneous ones*. 

The field in a dielectric is the superposition of the field E extr 
produced by the extraneous charges, and the field E b0Un ci of the 
bound charges. The resultant field is called microscopic (or true): 

E-rolcro “ E© x f r -4- E boun( i (2.7) 

The microscopic field changes greatly within the limits of the 
intermolecular distances. Owing to the motion of the bound charges, 
the field E ralcro also changes with time. These changes are not detect- 
ed in a macroscopic consideration. Therefore, a field is characterized 
by the quantity (2.7) averaged over an infinitely small (physically) 
volume, i.e. 

E = (E mloro ) = (Eextr) “b (E b ound) 

In the following, we shall designate the averaged field of the 
extraneous charges by E 0 , and the averaged field of the bound 
charges by E'. Accordingly, we shall define a macroscopic field as 
the quantity 

E = E 0 + E' (2.8) 

The polarization P is a macroscopic quantity. Therefore, E in 
Eq. (2.5) should be understood as the strength determined by 
Eq. (2.8). 

In the absence of dielectrics (i.e. in a “vacuum”), the macroscopic 
field is 

E = E 0 — (Eextf) 

It is exactly this quantity that is understood to be E in Eq. (1.117). 

If the extraneous charges are stationary, the field determined by 
Eq. (2.8) has the same properties as an electrostatic field in a vac- 
uum. In particular, it can be characterized with the aid of the 
potential <p related to the field strength (2.8) by Eqs. (1.41) and 
(1.45). 


2.4. Space and Surface Bound Charges 

When a dielectric is not polarized, the volume density p' and 
the surface density o' of the bound charges equal zero. Polarization 
causes the surface density, and in some cases also the volume density 
of the bound charges to become diSerent from zero. 


* It is customary practice to call such charges free . This name is extremely 
unsuccessful, however, because in a number of cases extraneous charges are not 
at all free. 


66 


Electricity and Magnetism 


Figure 2.1 shows schematically a polarized dielectric with non- 
polar (a) and polar (b) molecules. Inspection of the figure shows 
that the polarization is attended by the appearance of a surplus of 


00-00 

■ 

0 0J Gr 0 


-0-0~0~0 

1 

O- 0 01-0- 

_wC 

■OO'O'O" 


0 O' O' O' 

c 

oooo 

1 


•o- -a 0O- 



(a) (b) 


Fig. 2.1 


- 0 ' 


+ 0 ' 


bound charges of one sign in the thin surface layer of the dielectric* 
If the normal component of the field strength E for the given section 
of the surface is other than zero, then under the action of the field, 

charges of one sign will move away inward , 
and of the other sign will emerge. 

There is a simple relation between the 
polarization P and the surface density of 
the bound charges o'. To find it, let us 
consider an infinite plane-parallel plate 
of a homogeneous dielectric placed in 
a homogeneous electric field (Fig. 2.2). 
Let us mentally separate an elementary 
volume in the plate in the form of a very 
thin cylinder with generatrices parallel to 
E in the dielectric, and with bases of area 
AS coinciding with the surfaces of the 
plate. The magnitude of this volume is 
AV = IAS cos a 

where l = distance between the bases of the cylinder 

a = angle between the vector E and an outward normal to 
the positively charged surface of the dielectric. 

The volume AV has a dipole electric moment of the magnitude 

PAV = PIAS cos a 

(P is the magnitude of the polarization). 








Electric Field in Dielectrics 


67 


From the macroscopic viewpoint, the volume being considered 
is equivalent to a dipole formed by the charges + a'AS and — <j' AS 
with a spacing of Z. Therefore, its electric moment can be written 
in the form o'ASZ. Equating the two expressions for the electric 
moment, we get 

PIAS cos a = o'ASl 

Hence we get the required relation between o' and P: 

o' — P cos a — (2.9) 

where P n is the projection of the polarization onto an outward 
normal to the relevant surface. For the right-hand surface in Fig. 2.2 f 
we have P n >0, accordingly, o' for it is 
positive; for the left-hand surface P n < 0, 
accordingly, o' for it is negative. 

Expressing P through x and E by means 
of Eq. (2.5), we arrive at the formula 

o' = %* 0 E n (2.10) 

where E n is the normal component of the 
field strength inside the dielectric. According 
to Eq. (2.10), at the places where the field 
lines emerge from the dielectric (E^ >0), 
positive bound charges come up to the surface, while where the field 
lines enter the dielectric ( E n < 0), negative surface charges appear. 

Equations (2.9) and (2.10) also hold in the most general case when 
an inhomogeneous dielectric of an arbitrary shape is in an inhomo- 
geneous electric field. By P n and E n in this case, we must understand 
the normal component of the relevant vector taken in direct proxim- 
ity to the surface element for which o' is being determined. 

Now let us turn to finding the volume density of the bound charges 
appearing inside an inhomogeneous dielectric. Let us consider an 
imaginary small area AS (Fig. 2.3) in an inhomogeneous isotropic 
dielectric with non-polar molecules. Assume that a unit volume 
of the dielectric has n identical particles with a charge of ~\-e and n 
identical particles with a charge of — e. In close proximity to area 
AS , the electric field and the dielectric can be considered homo- 
geneous. Therefore, when the field is switched on, all the positive 
charges near AS will be displaced over the same distance Z x in the 
direction of E, and all the negative charges will be displaced in 
the opposite direction over the same distance Z 2 (see Fig. 2.3). A cer- 
tain number of charges of one sign (positive if a <C n/2 and negative 
if a >jx/2) will pass through area AS in the direction of a normal 
to it, and a certain number of charges of the opposite sign (negative 
if a ji/ 2 and positive if a >ji/ 2) in the direction opposite to n. 
Area AS will be intersected by all the charges -\-e that were at 





68 


Electricity and Magnetism 


a distance of not over l x cos a from it before the field was switched 
on, i.e. by all the +*’s in an oblique cylinder of volume l x AS cos a. 
The number of these charges is nl x AS cos a, while the charge they 
carry in the direction of a normal to the area is enl x AS cos a (when 
a > n/ 2, the charge carried in the direction of the normal as a result 
of displacement of the charges +e will be negative). Similarly, area 
AS will be intersected by all the charges — e in the volume l t AS cos a. 
These charges will carry a charge of enl 2 AS cos a in the direction 
of a normal to the area (inspection of Fig. 2.3 shows that when 
a <Z n/2, the charges — e will carry the charge — enl 2 AS cos a 
through AS in the direction opposite to n, which is equivalent 
to carrying the charge enl 2 AS cos a in the direction of n). 

Thus, when the field is switched on, the charge 

Aq' = enl i AS cos a + enl 2 AS cos a = en (l t + l z ) AS cos a 


is carried through area AS in the direction of a normal to it. The 
sum l x + l 2 is the distance l over which the positive and negative 
bound charges are displaced toward one another in the dielectric. 
As a result of this displacement, each pair of charges acquires the 
dipole moment p = el = e (l x + h)- The number of such pairs in 
a unit volume is n . Consequently, the product e (l x + l 2 ) n = 
= elrt = pn gives the magnitude of the polarization P . Thus, the 
charge passing through area AS in the direction of a normal to it 
when the field is switched on is [see Eq. (2.9)1 

!■' "7 A q' = PAS cos a 

Since the dielectric is isotropic, the directions of the vectors E 
and P coincide (see Fig. 2.3). Consequently, a is the angle between 
the vectors P and n, and in this connection we can write 

Ag' = PnA5 


Passing over from deltas to differentials, we get 

dq = Pn dS = P dS 

We have found the bound charge dq* that passes through elementary 
area dS in the direction of a normal to it when the field is switched 
on; P is the polarization set up under the action of the field at the 
location of area dS . 

Let us imagine closed surface S inside the dielectric. When the 
field is switched on, a bound charge q r will intersect this surface and 
emerge from it. This charge is 


gem — ^ dq — P dS 


(we have agreed to take the outward normal to area dS for closed 
surfaces). As a result, a surplus bound charge will appear in the 




Electric Field in Dielectrics 


69 


volume enclosed by surface S. Its value is 

ftury — ^em — — (V, P dS = — (2.11) 
8 

(Op is the flux of the vector P through surface 5). 



Fig. 2.4 


Introducing the volume density of the bound charges p\ we can 
write 

ffsUT^ ^ P dV 
v 

(the integral is taken over the volume enclosed by surface 5). We 
thus arrive at the formula 

j p' dV= — §FdS 

Let us transform the Surface integral according to the Ostrogradsky- 
Gauss theorem [see Eq. (1.108)1. The result is 

f p 'dF= — j VP dV 

V V 

This equation must be observed for any arbitrarily chosen vol- 
ume V . This is possible only if the following equation is observed 
at every point of the dielectric: 

p' = _ V P (2.12) 

Consequently, the density of bound charges equals the divergence 
of the polarization P taken with the opposite sign. 

We obtained Eq. (2.12) when considering a dielectric with non- 
polar molecules. This equation also holds, however, for dielectrics 
with polar molecules. 

Equation (2.12) can be given a graphical interpretation. Points 
with a positive VP are sources of the field of the vector P, and the 
lines of P diverge from them (Fig. 2.4). Points with a negative VP 


70 


Electricity and Magnetism 


are sinks of the field of the vector P, and the lines of P converge 
at them. In polarization of the dielectric, the positive bound charges 
are displaced in the direction of the vector P, i.e. in the direction of 
the lines P; the negative bound charges are displaced in the opposite 
direction (in the figure the bound charges belonging to separate 
molecules are encircled by ovals). As a result, a surplus of negative 
bound charges is formed at places with a positive VP, and a surplus 
of positive bound charges at places with a negative VP. 

Bound charges differ from extraneous ones only in that they can- 
not leave the confines of the molecules which they are in. Otherwise, 
they have the same properties as all other charges. In particular, 
they are sources of an electric field. Therefore, when the density 
of the bound charges p' differs from zero, Eq. (1.117) must be written 
in the form 

VE=~(p + p') (2.13) 

Here p is the density of the extraneous charges. 

Let us introduce Eq. (2.5) for P into Eq. (2.12) and use Eq. (1.103). 
The result is 

p' ® - V (X e oE) - - e 0 V (XE) - - e 0 (EV X + XVE) 

Substituting for vE its value from Eq. (2.13), we arrive at the equa- 
tion 

P ' = — e 0 EVx — XP — XP' 

Hence 

p'=- (eoEVx + XP) (2-14) 

We can see from Eq. (2.14) that the volume density of bound 
charges can differ from zero in two cases: (1) if a dielectric is not 
homogeneous (Vx ^ 0), and (2) if at a given place in a dielectric 
the density of the extraneous charges is other than zero (p ^ 0). 

When there are no extraneous charges in a dielectric, the volume 
density of the bound charges is 

p’ = T+x" evx (2.15) 

2.5. Electric Displacement Vector 

We noted in the preceding section that not only extraneous, but 
also bound charges are sources of a field. Accordingly, 

ve=-L( p +p') 

*0 


(see Eq. (2.13)1. 


(2.16) 


Electric Field in Dielectrics 


71 


Equation (2.16) is of virtually no use for finding the vector E 
because it expresses the properties of the unknown quantity E 
through bound charges, which in turn are determined by the unknown 
quantity E [see Eqs. (2.10) and (2.14)]. 

Calculation of the fields is often simplified if we introduce an aux- 
iliary quantity whose sources are only extraneous charges p. To 
establish what this quantity looks like, let us introduce Eq. (2.12) 
for p' into Eq. (2.16): 

VE = -±-(p-VP) 

whence it follows that 

V(e 0 E + P) =p (2.17) 

(we have put e 0 inside the del symbol). The expression in parenthe- 
ses in Eq. (2.17) is the required quantity. It is designated by the 
symbol D and is called the electric displacement (or electric induc- 
tion). 

Thus, the electric displacement is a quantity determined by the 
relation 

D = e 0 E + P (2.18) 

Inserting Eq. (2.5) for P, we get 

D = e 0 E + e 0 XE = e 0 (1 + %) E (2.19) 

The dimensionless quantity 

e = 1 + X (2.20) 

is called the relative permittivity or simply the permittivity of a me- 
dium*. Thus, Eq. (2.19) can be written in the form 

D = e 0 eE (2.21) 

According to Eq. (2.21), the vector D is proportional to the vector E. 
We remind our reader that we are dealing with isotropic dielectrics. 
In anisotropic dielectrics, the vectors E and D, generally speaking, 
are not collinear. 

In accordance with Eqs. (1.15) and (2.21), the electric displace- 
ment of the field of a point charge in a vacuum is 






( 2 . 22 ) 


The unit of electric displacement is the coulomb per square 
metre (C/m 2 ). 

Equation (2.17) can be written as 

VD = P (2.23) 

* The so-called absolute permittivity of a medium e a = e 0 6 is introduced in 
electrical engineering. This quantity is deprived of a physical meaning, however, 
and we shall not use it. ^ “ 


72 


Electricity and Magnetism 


Integration of this equation over the arbitrary volume V yields 

j VD dV — f p dV 

v v 

Let us transform the left-hand side according to the Ostrogradsky- 
Gauss theorem [see Eq. (1.108)]: 

■§DdS=fpdV (2.24) 

S V 

The quantity on the left-hand side is d> D —the flux of the vector D 
through closed surface S, while that on the right-hand side is the 
sum of the extraneous charges 2 9i enclosed by this surface* Hence, 
Eq. (2.24) can be written in the form 

<J> D = 2 (2-25) 

Equations (2.24) and (2.25) express Gauss’s theorem for the vec- 
tor D: the flux of the electric displacement through a closed surface 
equals the algebraic sum of the extraneous charges enclosed by this 
surface . 

In a vacuum, P = 0, so that the quantity D determined by 
Eq. (2.18) transforms into e 0 E, and Eqs. (2.24) and (2.25) transform 
into Eqs. (1.116) and (1.114). 

The unit of the flux of the electric displacement vector is the 
coulomb. By Eq. (2.25), a charge of 1 C sets up a displacement flux 
of 1 C through the surface surrounding it. 

The field of the vector D can be depicted with the aid of electric 
displacement lines (we shall call them displacement lines for brevi- 
ty’s sake). Their direction and density are determined in exactly 
the same way as for the lines of the vector E (see Sec. 1.5). The lines 
of the vector E can begin and terminate at both extraneous and 
bound charges. The sources of the field of the vector D are only 
extraneous charges. Hence, displacement lines can begin or termi- 
nate only at extraneous charges. These lines pass without interrup- 
tion through points at which bound charges are placed. 

The electric induction 5 * 1 in the Gaussian system is determined 
by the expression 

D = E + 4nP (2.26) 

Substituting for P in this equation its value from Eq* (2.6) f we get 

D = (1 + in X ) E (2.27) 

The quantity 

e = 1 + 4jtx (2.28) 




The term “electric displacement” is not applied to quantity (2.27), 




Electric Field in Dielectrics 


73 


is called the permittivity. Introducing this quantity into Eq. (2.27) 
we get 

D = eE (2.29) 

In the Gaussian system, the electric induction in a vacuum coin- 
cides with the field strength E. Consequently, the electric induction 
of the field of a point charge in a vacuum is determined by Eq. (1.16). 

By Eq. (2.22), the electric displacement set up by a charge of 1 C 
at a distance of 1 m is 


D 


JL JL 

4 n r % 


C/m 2 


In the Gaussian system, the electric induction in this case is 
Z> = -^ = -^^= 3xl ° s cgse D 

Thus, 

1 C/m 2 = 4n x 3 x 10 6 cgse D 

In the Gaussian system, the expressions of Gauss’s theorem 
have the form 

§DdS = 4jt j pdV (2.30) 

(2.31) 

According to Eq. (2.31), a charge of 1 C sets up a flux of the electric 
induction vector of Anq = An x 3 X 10® cgse^. The following 
relation thus exists between the units of flux of the vector D: 

1 C = An x 3 x 10 9 cgse<p D 


2.6. Examples of Calculating the Field 
in Dielectrics 

We shall consider several examples of fields in dielectrics to 
reveal the meaning of the quantities D and e. 

Field Inside a Flat Plate. Let us consider two infinite parallel 
oppositely charged planes. Let the field they produce in a vacuum 
be characterized by the strength E 0 and the displacement D 0 = 
= e 0 E 0 . Let us introduce into this field a plate of a homogeneous 
isotropic dielectric and arrange it as shown in Fig. 2.5. The dielectric 
becomes polarized under the action of the field, and bound charges 
of density a' will appear on its surfaces. These charges will set up 
a homogeneous field inside the plate whose strength by Eq. (1.121) 
is E‘ = a7e 0 . In the given case, E' = 0 outside the dielectric. 


74 


Electricity and Magnetism 


The field strength E 0 is a/e 0 . Both fields are directed toward each 
other, hence inside the dielectric we have 

£=£ 0 -£' = £ 0 --^ = -±-(a-a') (2.32) 

Outside the dielectric, E = E 0 . 

The polarization of the dielectric is due to field (2.32). The latter 
is perpendicular to the surfaces of the plate. Hence, E n — E y and 

in accordance with Eq. (2.10), or' = % E o^- 
Using this value in Eq. (2.32), we get 

E=E o—xE 

whence 

E ° E ° (2.33) 


+ 0- -cr* 





... 



+ 








+ 

+ 






+ 

+ 







- + 




6 

a 


& 

& So 
So 

e °> 


E = 


l + x c 


Thus, in the given case, the permittivity e 
shows how many times the field in a di- 
electric weakens. 

Multiplying Eq. (2.33) by e 0 e, we get the 
electric displacement inside the plate: 

D = e 0 eE = e 0 E 0 = D 0 (2.34) 

Hence, the electric displacement inside the 
plate coincides with that of the external 
field£> 0 . Substituting <x/e 0 for£ 0 inEq. (2.34), 
we find 

D - or (2.35) 

To find o', let us express E and E 0 in Eq. (2.33) through the charge 
densities: 

— (o — o') = — 
e 0 v ' e 0 e 


Fig. 2.5 


whence 


a = 


(2.36) 


Figure 2.5 has been drawn assuming that e = 3. Accordingly, the 
density of the field lines in the dielectric is one-third of that outside 
the plate. The lines are equally spaced because the field is homogene- 
ous. In the given case, o' can be found without resorting to Eq. (2.36). 
Indeed, since the field intensity inside the plate is one-third of that 
outside it, then of three field lines beginning (or terminating) on 
extraneous charges, two must terminate (or begin respectively) on 
bound charges. It thus follows that the density of the bound charges 
must be two-thirds that of the extraneous charges. 

In the Gaussian system, the field strength E' produced by the 
bound charges cr' is 4 jio'. Therefore, Eq. (2.32) becomes 

E=EQ—E' = E 0 —4no' 


(2.37) 



Electric Field in Dielectrics 


75 


The surface density a' is associated with the field strength E by the 
equation o' = %E„. We can thus write that 

E = E 0 — AnxE 


whence 


E 


g. _ T E t 

1+4jxx e 


Thus, the permittivity e, like its counterpart e in the SI, shows how 
many times the field inside a dielectric weakens. Therefore, the 



values of e in the SI and the Gaussian system coincide. Hence, taking 
into account Eqs. (2.20) and (2.28), we conclude that the susceptibil- 
ities in the Gaussian system (Xgs) and in the SI (xsr) differ from 
each other by the factor 4jt: 

Xsi = 4 jixgs (2.38) 

Field Inside a Spherical Layer. Let us surround a charged sphere 
of radius R with a concentric spherical layer of a homogeneous iso- 
tropic dielectric (Fig. 2.6). The bound charge q\ distributed with the 
density o' will appear on the internal surface of the layer (< 7 ' = 
= 4n/?jO'), and the charge q 2 distributed with the density o' 2 will 
appear on its external surface (q' 2 = 4 nR\<j' 2 ). The sign of the charge q\ 
coincides with that of the charge q of the sphere, while q\ has the 
opposite sign. The charges q\ and < 7 ' set up a field at a distance r 
exceeding R x and R 2 , respectively, that coincides with the field of 
a point charge of the same magnitude [see Eq. (1.124)1. The charges 
q[ and q' 2 produce no field inside the surfaces over which they are 


76 


Electricity and Magnetism 


distributed. Hence, the field strength E' inside a dielectric is 

E , =s _l gj _ 1 4nflfOt _ 1 itfoj 

4jte 0 r* 4neo r* e 0 r* 

and is opposite in direction to the field strength E 0 . The resultant 
field in a dielectric is 


2?(r) = 


E 0 —E' = 


1 

4ne 0 ' 


q_ 

r* 


1 Rjoj 

t> t r* 


(2.39) 


It diminishes in proportion to 1/r 4 . We can therefore state that 


EJR i) 
E(r) 


R\ 


i.e. Em-E(r)-^ 


where E (/? x ) is the field strength in a dielectric in direct proximity 
to the internal surface of the layer. It is exactly this strength that 
determines the quantity aj: 

oj = xt 0 E (i? t ) = %e 0 E (r) (2.40) 

(at each point of the surface | 2? n | = E). 

Introducing Eq. (2.40) into Eq. .(2.39), we get 


£(r) 


I g 
4ne 0 r 2 


1 i?f X e 0 £(r)r 2 
e 0 


= E 0 (r)- X E(r) 


From this equation, we find that inside a dielectric E = E 0 /e, and, 
consequently, D = e 0 2? 0 [compare with Eqs. (2.33) and (2.34)]. 

The field inside a dielectric changes in proportion to 1/r 2 . There- 
fore, the relation o' : o' = R\ : holds. Hence it follows that q[ = q 

Consequently, the fields set up by these charges at distances exceed- 
ing Ft 2 mutually destroy each other so that outside the spherical lay- 
er E' = 0 and E = E 0 . 

Assuming that = R and R 2 = oo, we arrive at the case of 
a charged sphere immersed in an infinite homogeneous and isotropic 
dielectric. The field strength outside such a sphere is 


E 


1 g_ 

4neo er 2 


(2.41) 


The strength of the field set up in an infinite dielectric by a point 
charge will be the same. 

Both examples considered above are characterized by the fact 
that the dielectric was homogeneous and isotropic, and the surfaces 
enclosing it coincided with the equipotential surfaces of the field of 
extraneous charges. The result we have obtained in these cases is 
a general one. If a homogeneous and isotropic dielectric completely fills 
the volume enclosed by equipotential surfaces of the field of extraneous 
charges , then the electric displacement vector coincides with the vector 
of the field strength of the extraneous, charges multiplied by e 0 , and. 



Electric Field in Dielectrics 


77 


therefore, the field strength inside the dielectric is Hz of that of the 
field strength of the extraneous charges . 

If the above conditions are not observed, the vectors D and e 0 E do 
not coincide. Figure 2.7 shows the field in the plate of a dielectric. 
The plate is skewed relative to the planes carrying extraneous charges. 
The vector E' is perpendicular to the faces of the plate, therefore 


+cr -0' +G f -o 



£ 



Fig. 2.8 


E and E 0 are not collinear. The vector D is directed the same as E, 
consequently D and e 0 E 0 do not coincide in direction. We can show 
that they also fail to coincide in magnitude. 

In the examples considered above owing to the specially selected 
shape of the dielectric, the field E' differed from zero only inside the 
dielectric. In the general case, E' may differ from zero outside the 
dielectric too. Let us place a rod made of a dielectric into an initial- 
ly homogeneous field (Fig. 2.8). Owing to polarization, bound charges 
of opposite signs are formed on the ends of the rod. Their field 
outside the rod is equivalent to the field of a dipole (the lines of E' 
are dash ones in the figure). It is easy to see that the resultant field E 
near the ends of the rod is greater than the field E 0 . 


2.7. Conditions on the Interface Between 
Two Dielectrics 

Near the interface between two dielectrics, the vectors E and D 
must comply with definite boundary conditions following from the 
relations (1.112) and (2.23): 

IVE] = 0, VD = p 

Let us consider the interface between two dielectrics with the 
permittivities e x and e, (Fig. 2.9). We choose an arbitrarily direct- 


78 


Electricity and Magnetism 


ed x-axis on this surface. We take a small rectangular contour of 
length a and width b that is partly in the first dielectric and partly 
in the second one. The x-axis passes through the middle of the 
sides 6. 

Assume that a field has been set up in the first dielectric whose 
strength is E lt and in the second one whose strength is E 2 . Since 
IVEl — 0, the circulation of the vector E around the contour we 



Fig. 2.9 


Fig. 2.10 


have chosen must equal zero [see Eq. (1.110)1. With small dimensions 
of the contour and the direction of circumvention shown in Fig. 2.9, 
the circulation of the vector E can be written in the form 

E,dl= E lx a — E 2> x a + (E b ) 2 b (2.42) 

where (E h ) is the mean value of E x on sections of the contour perpen- 
dicular to the interface. Equating this expression to zero, we arrive 
at the equation 

(E z , x — E u x) a = (Eb) 

In the limit, when the wid th j? of the contour tends to zero, we get 

E Ux = E 2 , x (2.43) 

The values of the projections of the vectors E x and E 2 onto the x-axis 
are taken in direct proximity to the interface between the boundary 
of the dielectrics. 

Equation (2.43) is obeyed when the x-axis is selected arbitrarily. 
It is only essential that this axis be in the plane of the interface 
between the dielectrics. Inspection of Eq. (2.43) shows that with 
such a selection of the x-axis when E 1%x = 0, the projection of E 2X 
will also equal zero. This signifies that the vectors Ej and E 2 at two 
close points taken at opposite sides of the interface are in the same 
plane as a normal to the interface. Let us represent each of the vectors 
E x and E 2 in the form of the sum of the normal and tangential com- 
ponents: 

E| = E lt n ■+■ E| t t ; E 2 = E 2t n + E 2f * 

In accordance with Eq. (2.43) 

E\ m x = E 2% x 


(2.44) 



Electric Field in Dielectrics 


79 


Here is the projection of the vector E* onto the unit vector t 
directed along the line of intersection of the dielectric interface with 
the plane containing the vectors E 2 and E 2 . 

Substituting in accordance with Eq. (2.21) the projections of the 
vector D divided by e 0 e for the projections of the vector E, we get 
the proportion 

^1. T __ ^2. X 

e 0 e, 

whence it follows that 

1 (2.45) 

U 2> T e 2 

Now let us take an imaginary cylindrical surface of height h on the 
interface between the dielectrics (Fig. 2.10). Base S x is in the first 
dielectric, and base S 2 in the second. Both bases are identical in size 
(S 1 = S 2 = S) and are so small that within the limits of each of 
them the field may be considered homogeneous. Let us apply Gauss’s 
theorem [see Eq. (2.25)] to this surface. If there are no extraneous 
charges on the interface between the dielectrics, the right-hand side 
in Eq. (2.25) equals zero. Hence, O d = 0. 

The flux through base is D l n S , where D l n is the projection of 
the vector D in the first dielectric onto the normal n v Similarly, the 
flux through base S 2 is D 2n S, where D 2 n is the projection of the 
vector D in the second dielectric onto the normal n 2 . The flux through 
the side surface can be written in the form ( D n )S side , where ( D n > is 
the value of D n averaged over the entire side surface, and S sl d e is the 
magnitude of this surface. We can thus write that 

C>£) = Z> lt n S + Z?2, + {Dn) ^Side = 0 (2.46) 

If the altitude h of the cylinder is made to tend to zero, then 5 slde 
will also tend to zero. Hence, in the limit, we get 

D\ , n = D 2t n 

Here Z> | 71 is the projection onto n f of the vector D in the i-th dielec- 
tric in direct proximity to its interface with the other dielectric. 
The signs of the projections are different because the normals n 1 
and n 2 to the bases of the cylinder have opposite directions. If we 
project D x and D 2 onto the same normal, we get the condition 

D un = D 2 , „ (2.47) 

Using Eq. (2.21) to replace the projections of D with the corre- 
sponding projections of the vector E multiplied by £ 0 e, we get the 
relation 

e 0 S i E it n = Co e 2^2, n 


n 

. n e l 


whence 


(2.48) 



80 


Electricity and Magnetism 


The results we have obtained signify that when passing through the 
interface between two dielectrics, the normal component of the vector 
D and the tangential component of the vector E change continu- 
ously. The tangential component of the vector D and the normal 
component of the vector E, however, are disrupted when passing 
through the interface. 

Equations (2.44), (2.45), (2.47), and (2.48) determine the condi- 
tions which the vectors E and D must comply with on the interface 
between two dielectrics (if there are no extraneous charges on this 




interface). We have obtained these equations for an electrostatic 
field. They also hold, however, for fields varying with time (see 
Sec. 16.3). 

The conditions we have found also hold for the interface between 
a dielectric and a vacuum. In this case, one of the permittivities must 
be taken equal to unity. 

We must note that condition (2.47) can be obtained on the basis 
of the fact that the displacement lines pass through the interface 
between two dielectrics without being interrupted (Fig. 2.11). Accord- 
ing to the rule for drawing these lines, the number of lines arriving 
at area A S from the first dielectric is D l AS 1 = D X AS cos a x . Simi- 
larly, the number of lines emerging from area AS into the second 
dielectric is D 2 AS 2 — D Z AS cos a 2 . If the lines are not interrupted 
at the interface, both these numbers must be the same: 

D X AS cos a t = D^AS cos a* 

Cancelling AS and taking into account that the product D cos a gives 
the value of the normal component of the vector D f we arrive at con- 



Electric Field in Dielectrics 


81 


The displacement lines are bent (refracted) on the interface be- 
tween dielectrics, owing to which the angle a between a normal to 
the interface and the line D changes. Inspection of Fig. 2.12 shows 
that 

tan a 4 : tan ctj = - P — — : - 

1 ^ D t , n D t , n 

whence with account taken of Eqs. (2.45) and (2.47), we get the law 
of displacement line refraction: 


tan a t jsj 

tan a, e. 


(2.49) 


W hen displacement lines pass into a dielectric with a lower permittiv- 
ity e, the angle made by them with a normal diminishes, hence, the 
lines are spaced farther apart; when the lines pass into a dielectric 
with a higher permittivity e, on the contrary, they become closer 
together. 


2.8. Forces Acting on a Charge 
in a Dielectric 

If we introduce into an electric field in a vacuum a charged body of 
such small dimensions that the external field within the body can be 
considered homogeneous, then the body will experience the force 

F - gE (2.50) 

To place a charged body in a field set up in a dielectric, a cavity 
must be made in the latter. In a fluid dielectric, the body itself forms 
the cavity by displacing the dielectric from the volume it occupies. 
The field inside the cavity E cav will differ from that in a continuous 
dielectric. Thus, we cannot calculate the force exerted on a charged 
body placed in a cavity as the product of the charge q and the field 
strength E in the dielectric before the body was introduced into it. 

When calculating the force acting on a charged body in a fluid 
dielectric, we must take another circumstance into account. Mechani- 
cal tension is set up on the boundary with the body in the dielectric. 
This sets up an additional mechanical force F ten acting on the body. 

Thus, the force acting on a charged body in a dielectric, generally 
speaking, cannot be determined by Eq. (2.50), and it is usually a very 
complicated task to calculate it. These calculations give an interest- 
ing result for a fluid dielectric. The resultant of the electric force 
gFcav and the mechanical force F ten is found to be exactly equal to 
gE, where E is the field strength in the continuous dielectric 

F = gE caT + F ten = gE 


(2.51) 


82 


Electricity and Magnetism 


The strength of the field produced in a homogeneous infinitely 
extending dielectric by a point charge is determined by Eq. (2.49). 
Hence, we get the following expression for the forces of interaction of 
two point charges immersed in a homogeneous infinitely extending 
dielectric: 


F = 


1 

4 ne 0 


Ci at 
er 1 


(2.52) 


This formula expresses Coulomb’s law for charges in a dielectric. 
It holds only for fluid dielectrics. 

Some authors characterize Eq. (2.52) as “the most general expres- 
sion of Coulomb’s law”. In this connection, we shall cite Richard 
P. Feynman: 

“Many older books on electricity start with the ‘fundamental* 
law that the force between two charges is... [Eq. (2.52) is given]..., a 
point of view which is thoroughly unsatisfactory. For one thing, it is 
not true in general; it is true only for a world filled with a liquid. 
Secondly, it depends on the fact that e is a constant which is only 
approximately true for most real materials.”* 

We shall not treat questions relating to the forces acting on a 
charge inside a cavity made in a solid dielectric. 


2.9. Ferroelectrics - 

There is a group of substances that can have the property of spon- 
taneous polarization in the absence of an external field. They are 
called ferroelectrics. This phenomenon was first discovered for 
Rochelle salt, and the first detailed investigation of the electrical 
properties of this salt was carried out by the Soviet physicists 

I. Kurchatov and P. Kobeko. 

Ferroelectrics differ from the other dielectrics in a number of 
features: 

1. Whereas the permittivity e of ordinary dielectrics is only sever- 
al units, reaching as an exception several scores (for example, for 
water 6 — 81), the permittivity of ferroelectrics may be of the order 
of several thousands. 

2. The dependence of P on E is not linear (see branch 1 of the 
curve shown in Fig. 2.13). Hence, the permittivity depends on the 
field strength. 

3. When the field changes, the values of the polarization P (and, 
therefore, of the displacement D too) lag behind the field strength E . 
As a result, P and D are determined not only by the value of E at 
the given moment, but also by the preceding values of E y i.e. they 


♦ R. P. Feynman, R. B. Leighton, M. Sands. The Feynman Lectures on Phys- 
ics . Vol. II. Reading, Mass., Addison- Wesley (1965), p. 10-8. 




Electric Field in Dielectrics 


83 


depend on the preceding history of the dielectric. This phenomenon is 
called hysteresis (from the Greek word “husterein” — to come late, 
be behind). Upon cyclic changes of the field, the dependence of P 
on E follows the curve shown in Fig. 2.13 and called a hysteresis loop. 
When the field is initially switched on, the polarization grows with E 
according to branch 1 of the curve. Diminishing of P takes place 
along branch 2. When E vanishes, the substance retains a value of 
the polarization P T called the residual polariza- 
tion. The polarization vanishes only under the 
action of an oppositely directed field E c . This 
value of the field strength is called the coer- 
cive force. Upon a further change in E y branch 
3 of the hysteresis loop is obtained, and so on. 

The behaviour of the polarization of fer- 
roelectrics is similar to that of the magneti- 
zation of ferromagnetics (see Sec. 7.9), and 
this is the origin of their name. 

Only crystalline substances having no centre 
of symmetry can be ferroelectrics. For exam- 
ple, the crystals of Rochelle salt belong to 
the rhombic system (see Sec. 13.2 of Vol. I, p. 369). The interaction 
of the particles in a ferroelectric crystal leads to the fact that their 
dipole moments line up spontaneously parallel to one another. In ex- 
clusive cases, the identical orientation of the dipole moments extends 
to the entire crystal. Ordinarily, however, regions appear in a crys- 
tal in whose confines the dipole moments are parallel to one another, 
but the directions of polarization in different regions are different. 
Thus, the resultant moment of an entire crystal may equal zero. The 
regions of spontaneous polarization are also called domains. Under 
the action of an external field, the moments of the domains rotate as 
a single whole, arranging themselves in the direction of the field. 

Every ferroelectric has a temperature at which the substance loses 
its unusual properties and becomes a normal dielectric. This temper- 
ature is called the Curie point. Rochelle salt has two Curie points, 
namely, — 15°Cand +22.5 °C, and it behaves like a ferroelectric only 
in the interval between these two temperatures. Its electrical 
properties are conventional at temperatures below — 15 °C and 
above +22.5 °C. 




CHAPTER 3 CONDUCTORS IN 

AN ELECTRIC FIELD 


3.1. Equilibrium of Charges 
on a Conductor 

The carriers of a charge in a conductor are capable of moving under 
the action of a vanishingly small force. Therefore, the following 
conditions must be observed for the equilibrium of charges on a con- 
ductor: 

1. The strength of the field everywhere inside the conductor must 
be zero: 

E = 0 (3.1) 

In Accordance with Eq. (1.41), this signifies that the potential inside 
the conductor must be constant (<p = const). 

2. The strength of the field on the surface of the conductor must be 
directed along a normal to the surface at every point: 

E = E n (3.2) 

Consequently, when the charges are in equilibrium, the surface of 
the conductor will be an equipotential one. 

If a charge q is imparted to a conducting body, the charge will be 
distributed so as to observe conditions of equilibrium. Let us imag- 
ine an arbitrary closed surface completely confined in a body. When 
the charges are in equilibrium, there is no field at every point inside 
the conductor; therefore the flux of the electric displacement vector 
through the surface vanishes. According to Gauss’s theorem, the sum 
of the charges inside the surface will also equal zero. This holds for 
a surface of any dimensions arbitrarily arranged inside a conductor. 
Hence, in equilibrium, there can be no surplus charges anywhere 
inside a conductor — they will all be distributed over the surface of 
the conductor with a certain density o. 

Since there are no surplus charges in a conductor in the state of 
equilibrium, the removal of substance from a volume taken inside 
the conductor will have no effect whatsoever on the equilibrium ar- 
rangement of the charges. Thus, a surplus charge will be distributed 
on a hollow conductor in the same way as on a solid one, i.e. along its 
external surface. No surplus charges can be located on the surface of 
a cavity in the state of equilibrium. This conclusion also follows from 
the fact that the like elementary charges forming the given charge q 
mutually repel one another and, consequently, tend to take up 
positions at the farthest distance apart. 



Conductors in an Electric Field 


85 


Imagine a small cylindrical surface formed by normals to the sur- 
face of a conductor and bases of the magnitude AS, one of which is 
inside and the other outside the conductor (Fig. 3.1). The flux of 
the electric displacement vector through the inner 
part of the surface equals zero because E and, 
consequently, D vanish inside the conductor. 

Outside the conductor in direct proximity to it, 
the field strength E is directed along a normal to 
the surface. Hence, for the side surface of the cyl- 
inder protruding outward, D n = 0, and for the 
outside base D n — D (the outside base is as- 
sumed to be very close to the surface of the con- 
ductor). Hence, the displacement flux through 
the surface being considered is D 4S, where D is 
the value of the displacement in direct proximity to the surface of the 
conductor. The cylinder contains an extraneous charge adS (a is 
the charge density at the given spot on the surface of the conductor). 




Applying Gauss’s theorem, we get D dS — <jdS y i.e. D = a. We thus 
see that the strength of the field near the surface of the conductor is 

— (3.3) 

e 0 e v ' 

where e is the permittivity of the medium surrounding the conductor 
[compare with Eq. (1.123) obtained for the case when e = 1]. 

Let us consider the field produced by the charged conductor shown 
in Fig. 3.2. At great distances from the conductor, equipotential 
surfaces have the shape of a sphere that is characteristic of a point 


86 


Electricity and Magnetism 


charge (owing to the lack of space, a spherical surface is shown in 
the figure at a small distance from the conductor; the dash lines are 
field lines). As we approach the conductor, the equipotential surfaces 
become more and more similar to the surface of the conductor, which 
is an equipotential one. Near the projections, the equipotential 
surfaces are denser, hence the field strength is also greater here. It 
thus follows that the density of the charges on the 
projections is especially great [see Eq. (3.3)1. We can 
arrive at the same conclusion by taking into account 
that owing to their mutual repulsion, charges tend to 
take up positions as far as possible from one another. 

Near depressions in a conductor, the equipoten- 
tial surfaces have a lower density (see Fig. 3.3). Ac- 
cordingly, the field strength and the density of the 
charges at these spots will be smaller. In general, the 
density of charges with a given potential of a conduc- 
tor is determined by the curvature of the surface — 
H grows with an increase in the positive curva- 
ture (convexity) and diminishes with an increase in 
the negative curvature (concavity). TheVlensity of charges is especi- 
ally high on sharp points. Consequently, the field strength near such 
points may be so great that the gas molecules surrounding the conduc- 
tor become ionized. Ions of the sign opposite to that of q are attract- 
ed to the conductor and neutralize its charge. Ions of the same sign 
as q begin to move away from the conductor, carrying along neutral 
molecules of the gas. The result is a noticeable motion of the gas 
called an electric wind. The charge of the conductor diminishes, it 
flows off the point, as it were, and is carried away* by the wind. This 
phenomenon is therefore called emanation of a charge from a point. 

3.2. A Conductor in an External 
Electric Field 

When an uncharged conductor is introduced into an electric field, 
the charge carriers come into motion: the positive ones in the direc- 
tion of the vector E, the negative ones in the opposite direction. As 
a result, charges of opposite signs called induced charges appear at 
the ends of the conductor (Fig. 3.4, the dash lines depict the external 
field lines). The field of these charges is directed oppositely to the 
external field. Hence, the accumulation of charges at the ends of 
a conductor leads to weakening of the field in it. The charge carriers 
will be redistributed until conditions (3.1) and (3.2) are observed, 
i.e. until the strength of the field inside the conductor vanishes and 
the field lines outside the conductor are perpendicular to its surface 
(see Fig. 3.4). Thus, a neutral conductor introduced into an electric 




Conductors in an Electric Field 


87 


field disrupts part of the field lines — they terminate on the negative 
induced charges and begin again on the positive ones. 

The induced charges distribute themselves over the outer surface 
of a conductor. If a conductor contains a cavity, then upon equi- 
librium distribution of the induced charges, the field inside it van- 
ishes. Electrostatic shielding is based on this phenomenon. If an 



Fig. 3.4 


instrument is to be protected from the action of external fields, it is 
surrounded by a conducting screen. The external field is compensated 
inside the screen by the induced charges appearing on its surface. 
Such a screen also functions quite well if it is made not solid, but 
in the form of a dense network. 


3.3. Capacitance 

A charge q imparted to a conductor distributes itself over its 
surface so that the strength of the field inside the conductor van- 
ishes. Such a distribution is the only possible one. Therefore, if we 
impart to a conductor already carrying the charge q another charge 
of the same magnitude, then the second charge must distribute itself 
over the conductor in exactly the same way as the first one. Other- 
wise, the charge will set up in the conductor a field differing from 
zero. We must note that this holds only for a conductor remote from 
other bodies (an isolated conductor). If other bodies are near the 
conductor, the imparting to the latter of a new portion of charge will 
produce either a change in the polarization of these bodies or a change 
in the induced charges on them. As a result, similarity in the dis- 
tribution of different portions of the charge will be violated. 



88 


Electricity and Magnetism 


Thus, charges differing in magnitude distribute themselves on an 
isolated conductor in a similar way (the ratio of the densities of the 
charge at two arbitrary points on the surface of the conductor with 
any magnitude of the charge will be the same). It thus follows that 
the potential of an isolated conductor is proportional to the charge 
on it. Indeed, an increase in the charge a certain number of times leads 
to an increase in the strength of the field at every point of the space 
surrounding the conductor the same number of times. Accordingly, 
the work needed for transferring a unit charge from infinity to the 
surface of a conductor, i.e. the potential of the conductor, grows the 
same number of times. Thus, for an isolated conductor 

q = Ci p (3.4) 

The constant of proportionality C between the potential and the 
charge is called the capacitance. From Eq. (3.4), we get 

c-f (3.5) 

In accordance with Eq. (3.5), the capacitance numerically equals the 
charge which when imparted to a conductor increases its potential 
by unity. 

Let us calculate the potential of a charged sphere of radius R. 
The potential difference and the field strength are related by 
Eq. (1.45). We can therefore find the potential of the sphere <p by 
integrating Eq. (2.41) over r from/? to oo (we assume that the poten- 
tial at infinity equals zero): 

< 3 - 6 > 

Comparing Eqs. (3.6) and (3.5), we find that the capacitance of an 
isolated sphere of radius R immersed in a homogeneous infinite 
dielectric of permittivity e is 

C = 4 m 0 sR (3.7) 

The unit of capacitance is the capacitance of a conductor whose 
potential changes by 1 V when a charge of 1 C is imparted to it. 
This unit of capacitance is called the farad (F). 

In the Gaussian system, the formula for the capacitance of an 
isolated sphere has the form 

C = eR (3.8) 

Since e is a dimensionless quantity, the capacitance determined by 
Eq. (3.8) has the dimension of length. The unit of capacitance is the 
capacitance of an isolated sphere with a radius of 1 cm in a vacuum. 
This unit of capacitance is called the centimetre. According to 




Conductors in an Electric Field 


89 


Eq. (3.5), 

1 F = -^r = -test c 8 9e c " ' 9 x 1°“ ^ (3.9) 

An isolated sphere having a radius of 9 X 10* m, i.e. a radius 
1500 times greater than that of the Earth, would have a capacitance 
of 1 F. We can thus see that the farad is a very great unit. For this 
reason, submultiples of a farad are used in practice— the millifarad 
(mF), the microfarad (pF), the nanofarad (nF), and the picofarad 
(pF)'(see Vol. I, Table 3.1, p. 82). 


3.4. Capacitors 

Isolated conductors have a small capacitance. Even a sphere of 
the Earth’s size has a capacitance of only 700 pF. Devices are needed 
in practice, however, that with a low potential relative to the sur- 
rounding bodies would accumulate charges of an appreciable magni- 
tude (i.e. would have a high charge “capacity”). Such devices, called 
capacitors, are based on the fact that the capacitance of a conductor 
grows when other bodies are brought close to it. This is due to the 
circumstance that under the action of the field set up by the charged 
conductor, induced (on a conductor) or bound (on a dielectric) char- 
ges appear on the body brought up to it. Charges of the sign opposite 
to that of the charge q of the conductor will be closer to the conductor 
than charges of the same sign as q and, consequently, will have a 
greater influence on its potential. Therefore, when a body is brought 
close to a charged conductor, the potential of the latter diminishes in 
absolute value. According to Eq. (3.5), this signifies an increase in 
the capacitance of the conductor. 

Capacitors are made in the form of two conductors placed close to 
each other. The conductors forming a capacitor are called its plates. 
To prevent external bodies from influencing the capacitance of 
a capacitor, the plates are shaped and arranged relative to each other 
so that the field set up by the charges accumulating on them is con- 
centrated inside the capacitor. This condition is satisfied (see 
Sec. 1.14) by two plates arranged close to each other, two coaxial 
cylinders, and two concentric spheres. Accordingly, parallel-plate 
(plane), cylindrical, and spherical capacitors are encountered. 
Since the field is confined inside a capacitor, the electric displacement 
lines begin on one plate and terminate on the other. Consequently, 
the extraneous charges produced on the plates have the same magni- 
tude and are opposite in sign. 

The basic characteristic of a capacitor is its capacitance, by which 
is meant a quantity proportional to the charge q and inversely pro- 


90 


Electricity and Magnetism 


portional to the potential difference between the plates: 

( 3 . 

The potential difference <p x — (p 2 is called the voltage across the 
relevant points*. We shall use the symbol U to designate the voltage. 
Hence, Eq. (3.10) can be written as follows: 


C = — q — 
<Pi — <P* 


c=ir (3.ii) 

Here U is the voltage across the plates. 

The capacitance of capacitors is measured in the same units as 
that of isolated conductors (see the preceding section). 

The magnitude of the capacitance is determined by the geometry 
of the capacitor (the shape and dimensions of the plates and their 
separation distance), and also by the dielectric properties of the 
medium filling the space between the plates. Let us find the equation 
for the capacitance of a parallel-plate capacitor. If the area of a plate 
is S and the charge on it is q, then the strength of the field between 
the plates is 



[see Eqs. (1.121) and (2.33); e is the permittivity of the medium 
filling the gap between the plates]. 

In accordance with Eq. (1.45), the potential difference between 
the plates is 

Ed = £ “ £ 

Hence for the capacitance of a parallel-plate capacitor, we get 

c = (3.12) 


where S — area of a plate 

d = separation distance of the plates 

* t> e = permittivity of the substance filling the gap. 

It must be noted that the accuracy of determining the capacitance 
of a real parallel-plate capacitor by Eq. (3.12) is the greater, the 
smaller is the separation distance d in comparison with the linear 
dimensions of the plates. 

It can be seen from Eq. (3.12) that the dimension of the electric 
constant e 0 equals the dimension of capacitance divided by that of 
length. Accordingly, e 0 is measured in farads per metre [see 
Eq. (1.12)]. 

* A more general definition of the quantity called voltage will be given in 
Sec. 5.3 [see Eq. (5.18)]. 



Conductors in an Electric Field 


91 


If we disregard the dispersion of the field near the plate edges, we 
can easily obtain the following equation for the capacitance of 
a cylindrical capacitor: 


r 2jte<,eZ 
~ In 


(3.13) 


where l = length of the capacitor 

and R 2 = radii of the internal and external plates. 

The accuracy of determining the capacitance of a real capacitor 
by Eq. (3.13) is the greater, the smaller is the separation distance 
of the plates d = R 2 — R 2 in comparison with l and R v 
The capacitance of a spherical capacitor is 

< 314 > 


where and fi 2 are the radii of the internal and external plates. 

Apart from the capacitance, every capacitor is characterized by 
the maximum voltage t/ max that may be applied across its plates 
without the danger of a breakdown. When this voltage is exceeded, 
a spark jumps across the space between the plates. The result is 
destruction of the dielectric and failure of the capacitor. 



CHAPTER 4 ENERGY OF 

AN ELECTRIC FIELD 


4.1. Energy of a Charged Conductor 

The charge q on a conductor can be considered as a system of point 
charges A q. In Sec. 1.7, we obtained the following expression for the 
energy of interaction of a system of charges (see Eq. (1.39)1: 

W v = ( 4 - 1 ) 

Here <pi is the potential set up by all the charges except q t at the 
point where the charge q t is. 

The surface of a conductor is equipotential. Therefore, the poten- 
tials of the points where the point charges A q are located are identi- 
cal and equal the potential <p of the conductor. Using Eq. (4.1), we 
get the following expression for the energy of a charged conductor 

W p = ± 3 <pA <7 = 4*2 A ? = 4-W (4.2) 

Taking into account Eq. (3.5), we can write that 

< 4 - 3 > 

Any of these expressions gives the energy of a charged conductor. 


4.2. Energy of a Charged Capacitor 

Assume that the potential of a capacitor plate carrying the charge 
+9 is <Pi» and that of a plate carrying the charge — q is <p 2 . Conse- 
quently, each of the elementary charges A q into which the charge + q 
can be divided is at a point with the potential q^, and each of the 
charges into which the charge — q can be divided is at a point with 
the potential <p t . By Eq. (4.1), the energy of such a system of 
charges is 

M7 P = 4'((* t ‘^) < Pi + ( — ?) < l > 2l = T'^ ( < * >1 — < P 2 > = 'F^ (4-4) 

Using Eq. (3.11), we can write three expressions for the energy of 
a charged capacitor: 

W -&-<<* CU* 

p 2 2 C 2 


(4.5) 



Energy of an Electric Field 


93 


Equation (4.5) differs from (4.3) only in containing U instead of q>. 

The expression for the potential energy permits us to find the force 
with which the plates of a parallel-plate capacitor attract each 
other. Let us assume that the separation distance of the plates can 
be changed. We shall associate the origin of the 
x-axis with the left-hand plate (Fig. 4.1). The coordi- 
nate x of the second plate will therefore determine 
the separation distance d of the plates. According to 
Eqs. (3.12) and (4.5), we have 

W SL-^ X 

2 C 2e 0 eS x 

Let us differentiate this expression with respect to x, 
assuming that the charge on the plates is constant 
(the capacitor is disconnected from a voltage source). 

As a result, we obtain the projection of the force exerted on the 
right-liand plate onto the x-axis: 

F = dW * = g* 

* dx 2ei #S 



Fig. 4.1 


The magnitude of this expression gives the force with which the plates 
attract each other: 




g* 

2eoe5 


(4.6) 


Now let us try to calculate the force of attraction between the 
plates of a parallel-plate capacitor as the product of the strength of 
the field produced by one of the plates and the charge concentrated 
on the other one. By Eq. (1.120), the strength of the field set up by 
one plate is 


E = 


u __ g 

2eq 2 EqS 


(4.7) 


A dielectric weakens the field in the space between the plates e 
times, but this occurs only inside the dielectric [see Eq. (2.33) and 
the related text]. The charges on the plates are outside the dielectric 
and are therefore acted upon by the field strength given by Eq. (4.7). 
Multiplying the charge of a plate q by this strength, we get the fol- 
lowing expression for the force: 


F' 


g* 

2eqS 


(4.8) 


Equations (4.6) and (4.8) do not coincide. The value of the force 
given by Eq. (4.6) obtained from the expression for the energy agrees 
with experimental data. The explanation is that apart from the 
‘‘electric” force given by Eq. (4.8), the plates experience mechanical 
forces from the side of the dielectric that tend to spread them apart 



94 


Electricity and Magnetism 


(see Sec. 2.8; we must note that we have in mind a fluid dielectric). 
There is a dispersed field at the edges of the plates whose magnitude 
diminishes with an increasing distance from the 
\ / edges (Fig. 4.2). The molecules of the dielectric have 
/\ a dipole moment and experience the action of a force 
\ A * A / pulling them into the region with the stronger field 
'y- — [see Eq. (1.62)]. The result is an increase in the pres- 
rlT-T; sure between the plates and the appearance of a force 

that weakens the force given by Eq. (4.8) e 
II — I times. 

I’l If a charged capacitor with an air gap is partially 

IIIII immersed in a liquid dielectric, the latter will be 

IIIII drawn into the space between the plates (Fig. 4.3). 

IIIII This phenomenon is explained as follows. The per- 

IIIII mittivity of air virtually equals unity. Consequent- 

IIIII ly, before the plates are immersed in the dielectric, 

mil we can consider that the capacitance of the capaci- 

« tor is C 0 = e 0 S/d. and its energy is W 0 = q 2 /2C 0 . 

AVll When the space between the plates is partially filled 

i V \ with the dielectric, the capacitor can be considered 
y V as two capacitors connected in parallel, one of 
X which has a plate area of xS (x is the relative part 

of the space filled with the liquid) and is filled with 

Fig. 4.2 a dielectric for which e >1, and the other has a 
plate area equal to (1 — x)S . In the parallel con- 
nection of capacitors, their capacitances are summated: 


/Vra 


Fig. 4.2 


C = Cj -f Cg 


8o5(1 — x) I 8o eSx _ 
d "T d ~ 


= C 0 + - g8 - (c 7 1)5 x; 


Since C >C 0 , the energy W — q 2 /2C will be smaller than W 0 (the 
charge q is assumed to be constant — the capacitor was disconnected 
from the voltage source before being immersed in 
the liquid). Hence, the filling of the space between 
the plates with the dielectric is profitable from 
the energy viewpoint. This is why the dielectric 
is drawn into the capacitor and its level in the 
space separating the plates rises. This, in turn, 
results in an increase in the potential energy of 
the dielectric in the field of forces of gravity. In 
the long run, the level of the dielectric in the 
space will establish itself at a certain height cor- 
responding to the minimum total energy (elec- Fig. 4.3 

trical and gravitational). The above phenomenos 
is similar to the capillary rise of a liquid in the narrow space 
between plates (see Sec. 14.5 of Vol. I, p. 385). 

The drawing of the dielectric into the space between plates can 
also be explained from the microscopic viewpoint. There is a non- 




Energy of an Electric Field 


95 


uniform field at the edges of the capacitor plates. The molecules of 
the dielectric have an intrinsic dipole moment or acquire it under 
the action of the field; therefore, they experience forces that tend to 
transfer them to the region of the strong field, i.e. into the capacitor. 
These forces cause the liquid to be drawn into the space between the 
plates until the electric forces exerted on the liquid at the plate 
edges will be balanced by the weight of the liquid column. 


4.3. Energy of an Electric Field 


The energy of a charged capacitor can be expressed through quan- 
tities characterizing the electric field in the space between the plates. 
Let us do this for a parallel-plate capacitor. Introducing expression 
(3.12) for the capacitance into the equation W v — C U 2 I2 [see Eq. (4.5)), 
we get 





The ratio U/d equals the strength of the field between the plates; 
the product Sd is the volume occupied by the field. Hence, 

W p = ^lv (4.9) 

The equation W v = q 2 /2C relates the energy of a capacitor to the 
charge on its plates, while Eq. (4.9) relates this energy to the field 
strength. It is logical to ask the question: where, after all, is the 
energy localized (i.e. concentrated), what is the carrier of the ener- 
gy-charges or a field? This question cannot be answered within the 
scope of electrostatics, which studies the fields of fixed charges that 
are constant in time. Constant fields and the charges producing them 
cannot exist separately from each other. Fields varying in time, how- 
ever, can exist independently of the charges producing them and 
propagate in space in the form of electromagnetic waves. Experi- 
ments show that electromagnetic waves transfer energy. In partic- 
ular, the energy due to which life exists on the Earth is supplied 
from the Sun by electromagnetic waves; the energy that causes a radio 
receiver to sound is carried from the transmitting station by electro- 
magnetic waves, etc. These facts make us acknowledge the circum- 
stance that the carrier of energy is a field. 

If a field is homogeneous (which is the case in a parallel-plate 
capacitor), the energy confined in it is distributed in space with 
a constant density w equal to the energy of the field divided by the 
volume it occupies. Inspection of Eq. (4.9) shows that the density 
of the energy of a field of strength E set up in a medium with the 
permittivity e is 

e 0 e£* 

2 


w = 


(4.10) 


96 


Electricity and Magnetism 


With account taken of Eq. (2.21), we can write Eq. (4.10) as follows: 


e,e£* _ ED _ D * 
2 ~ 2 ~ 2e 0 e 


(4.11) 


In an isotropic dielectric, the directions of the vectors E and D 
coincide. We can therefore write the equation for the energy density 
in the form 


w 


ED 

2 


Substituting for D in this equation its value from Eq. (2.18), we get 
the following expression for w: 


E(e 0 E + P) _ _SqE» EP 
2 2 ' 2 


(4.12) 


The first addend in this expression coincides with the energy density 
of the field E in a vacuum. The second addend, as we shall proceed 
to prove, is the energy spent for polarization of the dielectric. 

The polarization of a dielectric consists in that the charges con- 
tained in the molecules are displaced from their positions under the 
action of the electric field E. The work done to displace the charges q t 
over the distance dr* per unit volume of the dielectric is 

dA= 2 ?£ Edr 4 = Ed(S qi r t ) 

(we consider for simplicity’s sake that the field is homogeneous). 

According to Eq. (2.1), 2 9i r « equals the dipole moment of a unit 

v — 1 _ 

volume, i.e. the polarization of the dielectric P. Hence, 

dA = EdP (4.13) 

The vector P is related to the vector E by the expression P = xe<>E 
(see Eq. (2.5)1. Hence, dP = Using this value of dP in 

Eq. (4.13), we get the expression 

dA = xeoEdE = d(^l) = d(-^-) 

Finally, integration gives us the following expression for the work 
done to polarize a unit volume of the dielectric: 



which coincides with the second addend in Eq. (4.12). Thus, expres- 
sions (4.11), apart from the intrinsic energy of a field e q E?I2, include 
the energy EP/2 spent for the polarization of the dielectric when the 
field is set up. 



Energy of an Electric Field 


97 


Knowing the density of the field energy at every point, we can 
find the energy of the field confined in any volume V . For this purpose, 
we must calculate the integral 



(4.15) 


Let us calculate as an example the energy of the field of a charged 
conducting sphere of radius R placed in a homogeneous infinite 
dielectric. The field strength here is a function only of r: 

E=— — 

4n«o er* 


Let us divide the space surrounding our sphere into concentric spher- 
ical layers of thickness dr. The volume of a layer is dV = 4 nr* dr. 
It contains the energy 


ilV = wiV =l^(^-^Y4nr>dr 


1 q* dr 

2 4ne 0 e r* 


The energy of the field is 



dW = 


2 


g* 

471608 



1 q* q* 

2 4ne 0 e/f 2 C 


[according to Eq. (3.7), 4jte 0 e/? is the capacitance of a sphere!. 

The expression we have obtained coincides with that for the energy 
of a conductor having the capacitance C and carrying the charge q 
[see Eq. (4.3)1. 


CHAPTER 5 STEADY ELECTRIC 

CURRENT 


5.4. Electric Current 


If a total charge other than zero is carried through an imaginary 
surface, an electric current (or simply a current) is said to flow 
through this surface. A current can flow in solids (metals, semiconduc- 
tors), liquids (electrolytes), and in gases (the flow of a current through 
a gas is called a gas discharge). 

For a current to flow, the given body (or given medium) must 
contain charged particles that can move within the limits of the 
entire body. Such particles are called current carri ers* The latter 
may be electrons, or ions, or, finally, macroscopic particles carrying 
a surplus charge (for example, charged dust particles and droplets). 

A current is produced if there is an electric field inside a body. 
The charge carriers participate in the molecular thermal motion and, 
consequently, travel with a certain velocity v even in the absence of 
a field. But in this case, an identical number of carriers of either 
sign pass on the average in both directions through an arbitrary area 
mentally drawn in the body, so that the current is zero. When a field 
is switched on, ordered motion with the velocity u is superposed onto 
the chaotic motion of the carriers with the velocity v*. The velocity 
of the carriers will thus be y + u. Since the mean value of v (but not 
of v) equals zero, then the mean velocity of the carriers is (u): 

{v + u> (▼) + = <u> 


It follows from what has been said above that an electric current can 
be defined as the ordered motion of electric charges. 

A quantitative characteristic of an electric current is the magnitude 
of the charge carried through the surface being considered in unit 
time. It is called the current strength, or more often simply the 
current. We must note that a current is in essence a flow of a charge 
through a surface (compare with the flow of a fluid, energy flux, etc.). 

If the charge dq is carried through a surface during the time dt, 
then the current is 




f) Url) 


r *4 
' dt 


(5.1) 


• Similarly, in a gas flow, ordered motion is superposed onto the chaotie 
thermal motion of the molecules. 



Steady Electric Current 


99 


An electric current may be produced by the motion of either positive 
or negative charges. The transfer of a negative charge in one direc- 
tion is equivalent to the transfer of a positive charge of the same mag- 
nitude in the opposite direction. If a current is produced by carriers 
of both signs, the positive carriers transferring the charge dq+ in one 
direction through the given surface during the time dt , and the nega- 
tive carriers the charge dq~ in the opposite direction during the same 
time, then 

r \dg~\ 

1 dt "T dt 


The direction of motion of the positive carriers has been conven- 
tionally assumed to be the direction of a current. 

A current may be distributed non-uniformly over the surface 
through which it is flowing. A current can be characterized in great- 
er detail by means of the current density vector j. This vector numer- 
ically equals the current dl through the area dS x arranged at the 
given point perpendicular to the direction of motion of the carriers 
divided by the magnitude of this area: 



The direction of j is taken as that of the velocity vector u + of the 
ordered motion of the positive carriers (or as the direction opposite 
to that of the vector u"). 

The field of the current density vector can be depicted by means 
of current lines that are constructed in the same way as the stream- 
lines in a flowing liquid, the lines of the vector E, etc. 

Knowing the current density vector at every point of space, we 
can find the current / through any surface S: 


/ = 


i 


jdS 


(5.3) 


It can be seen from Eq. (5.3) that the current is the flux of the current 
density vector through a surface [see Eq. (1.74)1. 

Assume that a unit volume contains n + positive carriers and /i“ 
negative ones. The algebraic value of the carrier charges is e * and e~ y 
respectively. If the carriers acquire the average velocities u* and u~ 
under the action of the field, then n + u + positive carriers will pass in 
unit time through unit area*, and they will transfer the charge 


• The expression for the number of molecules flying in unit time through unit 
area contains, in addition, the factor 1/4 due to the fact that the molecules move 
chaotically [see Eq. (11.23) of Vol. I, p. 303]. This factor is not present in the 
given case because all the carriers of a given sign have ordered motion in one di- 
rection. 



100 


Electricity and Magnetism 


eVu + . Similarly, the negative carriers will transfer the charge 
in the opposite direction. We thus get the following expres- 
sion for the current density: 

7 = e*n*u* + | e~ \ n~u~ (5.4) 

This expression can be given a vector form: 

j = e*n*n* + e~n~ u“ (5.5) 

(both addends have the same direction: the vector u~ is directed 
oppositely to the vector j; when it is multiplied by the negative 
scalar e~, we get a vector of the same direction as j). 

The product e*n* gives the charge density of the positive carriers p\ 
Similarly, e~n~ gives the charge density of the negative carriers p - . 
Hence, Eq. (5.5) can he written in the form 

j =p + u + + p-u- (5.6) 

A current that does not change with time is called steady (do not 
confuse with a direct current whose direction is constant, but whose 
magnitude may vary). For a steady current, we have 

(5.7) 

where q is the charge carried through the surface being considered 
during the finite time t. 

In the SI, the unit of current, the ampere (A), is a basic one. Its 
definition will be given on a later page (see Sec. 6.1). The unit of 
charge, the coulomb (C), is defined as the charge carried in one second 
through the cross section of a conductor at a current of one ampere. 

The unit of current in the cgse system is the current at which one 
cgse unit of charge (1 cgse„) is carried through a given surface in one 
second. From Eqs. (5.7) and (1.8) we find that 

1A = 3 X 10* cgsoj (5.8) 


5.2. Continuity Equation 

Let us consider an imaginary closed surface S (Fig. 5.1) in a medi- 
um in which a current is flowing. The expression j dS gives the 

s 

charge emerging in a unit time from the volume V enclosed by sur- 
face S. Owing to charge conservation, this quantity must equal the 
rate of diminishing of the charge q contained in the given volume: 

iq 


Steady Electric Current 


101 


Substituting for q its value j p dF*, we get the expression 

v 

§jdS=— g j pdV=- \%dV (5.9) 

S V V 

We have written the partial derivative of p with respect to t inside 
the integral because the charge density may depend not only on time, 

but also on the coordinates (the integral f p dV is a function only of 




time). Let us transform the left-hand side of Eq. (5.9) in accordance 
with the Ostrogradsky-Gauss theorem. As a result, we get 

j VjtfF=- j-g-dV (5.10) 

Equation (5.10) must be observed upon an arbitrary choice of the 

volume V over which the integrals are taken. This is possible only if 
at every point of space the condition is observed that 

vi=— -g (5.11) 

Equation (5.11) is known as the continuity equation. It [like Eq. 
(5.9)] expresses the law of charge conservation. According to Eq. (5.11), 
the charge diminishes at points that are sources of the vector j. 

For a steady current, the potential at different points, the charge 
density, and other quantities are constant. Hence, for a steady cur* 
rent, Eq. (5.11) has the form 

VJ = 0 (5.12) 

Thus, for a steady current, the vector j has no sources. This signifies 
that the current lines begin nowhere and terminate nowhere. Hence, 

the lines of a steady current are always closed. Accordingly, ^ J dS 

equals zero. Therefore, for a steady current, the picture similar to 
that shown in Fig. 5.1 has the form shown in Fig. 5.2. 



102 


Electricity and Magnetism 


5.3. Electromotive Force 

If an electric field is set up in a conductor and no measures are 
taken to maintain it, then motion of the current carriers will lead 
very rapidly to vanishing of the field inside the conductor and stop- 
ping of the current. To maintain a current for a sufficiently long time, 
it is necessary to continuously remove from the end of the conductor 
with the lower potential (the current carriers are assumed to be posi- 
tive) the charges carried to it by the current, and continuously sup- 
ply them to the end with the higher potential (Fig. 5.3). In other 

®-»- ©-»- 

'V 

^— © 

Fig. 5.3 

words, it is necessary to circulate the charges along a closed path. 
This agrees with the fact that the lines of a steady current are closed 
(see the preceding section). 

The circulation of the strength vector of an electrostatic field 
equals zero. Therefore, in a closed circuit in addition to sections on 
which the positive carriers travel in the direction of a decrease in 
the potential q>, there must be sections on which the positive charges 
are carried in the direction of a growth in <p, i.e. against the forces of 
the electrostatic field (see the part of the circuit in Fig. 5.3 shown 
by the dash line). Motion of the carriers on these sections is possible 
only with the aid of forces of a non-electrostatic origin, called 
extraneous forces. Thus, to maintain a current, extraneous forces are 
needed that act either over the entire length of the circuit or on 
separate sections of it. These forces may be due to chemical process- 
es, the diffusion of the current carriers in a non-uniform medium or 
through the interface between two different substances, to electric 
(but not electrostatic) fields set up by magnetic fields varying with 
time (see Sec. 9.1), etc. 

Extraneous forces can be characterized by the work they do on 
charges travelling along a circuit. The quantity equal to the work 
done by the extraneous forces on a unit positive charge is called the 
electromotive force (e.m.f.) % acting in a circuit or on a section of it. 
Hence, if the work of the extraneous forces on the charge q is A, then 



9 



(5.13) 



Steady Electric Current 103 


A comparison of Eqs. (5.13) and (1.31) shows that the dimension 
of the e.m.f. coincides with that of the potential. Therefore, % is 
measured in the same units as <p. 

The extraneous force F ex tr acting on the charge q can be represent- 
ed in the form 

Fextr = E*q (5.14) 

The vector quantity E* is called the strength of the extraneous force 
field. The work [of the extraneous forces on the charge q on circuit 
section 1-2 is 

2 2 

A a = j F extr dl = ? J E*dl 
1 1 

Dividing this work by q, we get the e.m.f. acting on the given section: 

2 

g 12 =jE*dl (5.15) 

1 

A similar integral calculated for a closed circuit gives the e.m.f. 
acting in this circuit: 

&=§E*dl (5.16) 

Thus, the e.m.f. acting in a closed circuit can be determined as the 
circulation of the strength vector of the extraneous forces. 

In addition to extraneous forces, a charge experiences the forces of 
an electrostati c field F^ = qE. Hence, the resultant force acting at 
each point of a ^circuit on the charge q is 

F = F* + F cxtr = q(E + E*) 

The work done by this force on the charge q on circuit section 1-2 is 
determined by the expression 
2 2 

A a = q j Edl-f? j E*dl = ?(<p,— q> 2 ) + ?£« (5.17) 

1 1 

The quantity numerically equal to the work done by the electrostat- 
ic and extraneous forces in moving a unit positive charge is defined 
as the voltage drop or simply the voltage U on the given section of 
the circuit. According to Eq. (5.17), 

— 92+^12 (5.18) 

A section of a circuit on which no extraneous forces act is called 
homogeneous. A section on which the current carriers experience 
extraneous forces is called inhomogeneous. For a homogeneous sec- 


104 


Electricity and Magnetism 


tion of a circuit 

= <Pi — ^2 (5,19) 

i.e. the voltage coincides with the potential difference across the 
ends of the section. 


5.4. Ohm’s Law. Resistance of Conductors 

The German physicist Georg Ohm (1789-1854) experimentally 
established a law according to which the current flowing in a homoge- 
neous (in the meaning that no extraneous forces are present) metal con- 
ductor is proportional to the voltage drop U 
in the conductor : 

I = -T U < 5 - 20 ) 

We remind our reader that fojr a homogeneous 
conductor the voltage U coincides with the 
potential difference q> x — <p a [see Eq. (5.18)1. 

The quantity designated by the symbol R in Eq. (5.20) is called 
the electrical resistance of a conductor. The unit of resistance is the 
ohm (Q) equal to the resistance of a conductor in which a current of 
1 A flows at a voltage of 1 V. 

The value of the resistance depends on the shape and dimensions 
of a conductor and also on the properties of the material it is made of. 
For a homogeneous cylindrical conductor 

* = p4 (5.21) 

where l = length of the conductor 
S = its cross-sectional area 

p s coefficient depending on the properties of the material 
and called the resistivity of the substance. 

If l = 1 and 5 = 1, then R numerically equals p. In the SI, p is 
measured in ohm-metres (£2-m). 

Let us find the relation between the vectors j and E at the same 
point of a conductor. In an isotropic conductor, the ordered motion 
of the current carriers takes place in the direction of the vector E. 
Therefore, the directions of the vectors j and E coincide*. Let us 
mentally separate an elementary cylindrical volume with generatri- 
ces parallel to the vectors j and E in the vicinity of a certain point 
(Fig. 5.4). A current equal to j dS flows through the cross section of the 
cylinder. The voltage across the cylinder is E dl , where E is the field- 


* In anisotropic bodies, the directions of the vectors] and E y generally speak- 
ing, do not coincide. The relation between j and E for such bodies is achieved 
with the aid of the conductance tensor. 


dl 






Fig. 5.4 



Steady Electric Current 


105 


strength at the given point. Finally, the resistance of the cylinder, 
according to Eq. (5.21), is p (dl/dS). Using these values in Eq. (5.20), 
we arrive at the equation 

jdS = 4%rEdl or ; = — £ 

pal p 

Taking advantage of the fact that the vectors j and E have the same 
direction, we can write 

i = -lE = oE (5.22) 

This equation expresses OhnTs law in the differential form. 

The quantity a in Eq. (5.22) that is the reciprocal of p is called the 
conductivity of a material. The unit that is the reciprocal of the 
ohm is called the siemens (S). The unit 
of o is accordingly the siemens per me* 
tre (S/m). 

Let us assume for simplicity’s sake 
that a conductor contains carriers of 
only one sign. According to Eq. (5.5), 
the current density in this case is 

j = enu (5.23) 

A comparison of this expression with 
Eq. (5.22) leads us to the conclusion 
that the velocity of ordered motion of 
current carriers is proportional to the 
field strength E, i.e. to the force impart- 
ing ordered motion to the carriers. Proportionality of the velocity to 
the force applied to a body is observed when apart from the force pro- 
ducing the motion, the body experiences the force of resistance of the 
medium. This force is due to the interaction of the current carriers 
with the particles which the substance of the conductor is built of. 
The presence of the force of resistance to ordered motion of the current 
carriers results in the electrical resistance of a conductor. 

The ability of a substance to conduct an electric current is charac- 
terized by its resistivity p or conductivity <j. Their magnitude is 
determined by the chemical nature of the substance and the sur- 
rounding conditions, in particular the ambient temperature. 

The resistivity p varies directly with the absolute temperature T 
for most metals at temperatures close to room one: 

p oc T (5.24) 

Deviations from this proportion are observed at low temperatures 
(Fig. 5.5). The dependence of p on T usually follows curve 1. The 
magnitude of the residual resistivity p re8 depends very greatly on 
the purity of the material and the presence of residual mechanical 



106 


Electricity and Magnetism 


stresses in the specimen. This is why p re8 appreciably diminishes 
after annealing. The resistivity p of a perfectly pure metal with an 
ideal regular crystal lattice vanishes at absolute zero. 

The resistance of a large group of metals and alloys at a temperature 
of the order of several kelvins vanishes in a jump (curve# in Fig. 5.5). 
This phenomenon, called superconductivity, was first discovered in 
1911 by the Dutch scientist Heike Kamerlingh Onnes (1853-1926) 
for mercury. Superconductivity was later discovered in lead, tin, 
zinc, aluminium, and other metals, as well as in a number of alloys. 
Every superconductor has its own critical temperature T cr at which 
it passes over into a superconducting state. The superconducting 
state is violated when a magnetic field acts on a superconductor. The 
magnitude of the critical field B cr (the symbol B stands for the 
magnetic induction — see Sec. 6.2) destroying superconductivity 
equals zero when T = T cr and grows with lowering of the tem- 
perature. 

A complete theoretical substantiation of superconductivity was 
given in 1957 by J. Bardeen, L. Cooper, and J. Schrieffer (see Vol. Ill, 
Sec. 8.2). 

The temperature dependence of resistance underlies the design of 
resistance thermometers. Such a thermometer is a metal (usually 
platinum) wire wound onto a porcelain or mica body. A resistance 
thermometer graduated according to constant temperature points 
makes it possible to measure both low and high temperatures with 
an accuracy of the order of several hundredths of a kelvin. Recent 
times have seen semiconductor resistance thermometers coming into 
greater and greater favour. 


5.5. Ohm’s Law for 

an Inhomogeneous Circuit Section 

The extraneous forces eE * act on current carriers on an inhomo- 
geneous section of a circuit in addition to the electrostatic forces eE . 
Extraneous forces are capable of producing ordered motion of current 
carriers to the same extent as electrostatic forces are. We found in 
the preceding section that in a homogeneous conductor the average 
velocity of ordered motion of the current carriers is proportional to 
the electrostatic force eE. It is quite obvious that where extraneous 
forces are exerted on the carriers in addition to the electrostatic 
forces, the average velocity of ordered motion of the carriers will 
be proportional to the total force eE + tfE*. Accordingly, the current 
density at these points is proportional to the sum of the strengths 
E + E*: 


j = <x(E + E*) 


(5.25) 



Steady Electric Current 


107 


Equation (5.25) summarizes Eq. (5.22) for an inhomogeneous con- 
ductor. It expresses Ohm’s law for an inhomogeneous section of 
a circuit in the differential form. 

We can pass over from Ohm’s law in the differential form to its 
integral form. Let us consider an inhomogeneous section of a circuit. 
Assume that there is a line inside this section (we shall call it the 
current path) complying with the following conditions: (1) in every 
cross section perpendicular to the path, the 
quantities j, a, E, E* have the same values 
with sufficient accuracy, and (2) the vectors j, 

E, and E* at every point are directed along a 
tangent to the path. The cross section of the 
conductor may vary (Fig. 5.6). t 

Let us choose an arbitrary direction of mo- 
tion along the path. Assume that the chosen 
direction corresponds to motion from end 1 
to end 2 of the circuit section (direction 1-2). Let us project the vec- 
tors in Eq. (5.25) onto the path element <21. The result is 

j^oiEi + Ef) (5.26) 

Owing to our assumption, the projection of each of the vectors equals 
the magnitude of the vector taken with the sign plus or minus depend- 
ing on the direction of the vector relative to <21. For example, j t = j 
if the current flows in direction 1-2, and U — — 7 if if flows in 
direction 2-1 . 

Owing to charge conservation, the steady current in each section 
must be the same. Therefore, the quantity / = j x S is constant along 
the path. The current in this case should be treated as an algebraic 
quantity. We remind our reader that we have chosen direction 1-2 
arbitrarily. Hence, if the current flows in the chosen direction, it 
should be considered positive, and if it flows in the opposite direction 
(i.e. from end 2 to end 7), it should be considered negative. 

Let us substitute the ratio I/S for j t and the resistivity p for the 
conductivity a in Eq. (5.26). We get 

f£- = E,+ E7 

Multiplication of the above equation by dl and integration along 
the path yield 

2 2 2 
/ f p-f-= f Eidl+ j Etdl 

ill 

The quantity p dl/S is the resistance of the path section of length 
dl, and the integral of this quantity is the resistance It of the cir- 
cuit section. The first integral in the right-hand side gives <p x — <p„ 



108 


Electricity and Magnetism 


and the second integral gives the e.m.f. acting on the section. 
We thus arrive at the equation 

= <p 2 + S« (5.27) 

The e.m.f. j? 12 , like the current I, is an algebraic quantity. When 
the e.m.f. facilitates the motion of the positive current carriers in 
the selected direction (in direction 1-2), we have % 1Z >0. If the 
e.m.f. prevents the motion of the positive carriers in the given direc- 
tion, t l2 < 0. 

Let us write Eq. (5.27) in the form 

/== il _-?»+ j L , (5.28) 

This equation expresses Ohm’s law for an inhomogeneous circuit 
section. Assuming that q) x = <p 2 » we get the equation of Ohm’s law 
for a closed circuit: 

/ = -§- (5.29) 

Here % = e.m.f. acting in the circuit 

R = total resistance of the entire circuit. 


5.6. Multiloop Circuits. 
KirchhofTs Rules 


The calculation of multiloop circuits or networks is considerably 
simplified if we use two rules formulated by the German physicist 
Gustav Kirchhoff (1824-1887). The first of them relates to the junc- 
tions of a circuit. A junction is defined as a point where three or 
more conductors meet (Fig. 5.7). A current flowing J 
toward a junction is considered to have one sign 
(plus or minus), and a current flowing out of a 
junction is considered to have the opposite sign 
(minus or plus). 

Kirchhoff ’s first rule, also called the junction^ 
rule, states that the algebraic sum of all the cur- 
rents coming into a function must be zero : 

27* = 0 (5.30) 



This rule follows from the continuity equation, i.e. in the long run 
from the law of charge conservation. For a steady current, vj equals 
zero everywhere [see Eq. (5.12)]. Hence, the flux of the vector j, 
i.e. the algebraic sum of the currents flowing through an imaginary 
closed surface surrounding a junction, must be zero. 



Steady Electric Current 


109 


Equation (5.30) can be written for each of the N junctions of 
a circuit. Only N — 1 equations will be independent, however, 
whereas the AT-th one will be a corollary of them. 

The second rule relates to any closed loop separated from a mul- 
tiloop circuit (see, for example, loop 1-2-3-4-1 in Fig. 5.8). Let us 



Fig. 5.8 

choose a direction of circumvention (for example, clockwise as in the 
figure) and apply Ohm’s law to each unbranched loop section: 

1 2^2 = ?2 — ?3 4- 
1 8^3 = <Ps — <P4 + 

/ 4 i?4 = q>4 — <Pi + ^4 


When these expressions are summated, the potentials can be can- 
celled, and we get the equation 

(5.31) 

that expresses Kirchhoff’s second rule, also called the loop rule. 

Equation (5.31) can be written for all the closed loops that can 
be separated mentally in a given multiloop circuit. Only the equa- 
tions for the loops that cannot be obtained by the superposition of 
other loops on one- another will be independent, however. For exam- 
ple, for the circuit depicted in Fig. 5.9, we can write three equations: 

(1) for loop 1-2-3 -6-1, 

(2) for loop 3-4-5-S-3 , and 

(3) for loop 1-2-3-4-5-6-1 . 

The last loop is obtained by superposition of the first two. The 
equations will therefore not be independent. We can take any two 
equations of the three as independent ones. 



110 


Electricity and Magnetism 


In writing the equations of the loop rule, we must appoint the 
signs of the currents and e.m.f.’s in accordance with the chosen direc- 
tion of circumvention. For example, the current I x in Fig. 5.9 must 
be considered negative because it flows oppositely to the chosen direc- 
tion of circumvention. The e.m.f. £*must also be considered negative 
because it acts in the direction opposite to that of circumvention, 
and so on. 

We may choose the direction of circumvention in each loop abso- 
lutely arbitrarily and independently of the choice of the directions in 
the other loops. It may happen here that the same current or the same 

e.m.f. may be included in different 
equations with opposite signs (this 
happens with the current / 2 in 
Fig. 5.9 for the indicated directions 
of circumvention in the loops). This 
is of no significance, however, be- 
cause a change in the direction 
around a loop results only in a re- 
versal of all the signs in Eq. (5.31). 

In compiling the equations, re- 
member that the same current flows 
in any cross section of an unbranched 
part of a circuit. For example, the 
same current / 2 flows from junction 
6 to the current source % 2 as from 
the source % 2 to junction 3. 

The number of independent equations compiled in accordance with 
the junction and loop rules equals the number of different currents 
flowing in a multiloop circuit. Therefore, if we know the e.m.f.’s and 
resistances for all the unbranched sections, we can calculate all the 
currents. We can also solve other problems, for instance find the 
e.m.f.’s that must be connected in each of the sections of a circuit 
to obtain the required currents with the given resistances. 



5.7. Power of a Current 

Let us consider an arbitrary section of a steady current circuit 
across whose ends the voltage U is applied. The charge q = It will 
flow during the time t through every cross section of the conductor. 
This is equivalent to the fact that the charge It is carried during the 
time t from one end of the conductor to the other. The forces of the 
electrostatic field and the extraneous forces acting on the given sec- 
tion do the work 

A — Uo — Ult (5.321 


Steady Electric Current 


111 


[we remind our reader that the voltage U is determined as the work 
done by the electrostatic and extraneous forces in moving a unit posi- 
tive charge; see Eq. (5.18)1. 

Dividing the work A by the time t during which it is done, we get 
the power developed by the current on the circuit section being: 
considered: 


P — UI — (<pi W 2 ) I + $ 12 -^ (5.33) 

This power may be spent for the work done by the circuit section 
being considered on external bodies (for this purpose the section 
must move in space), for the proceeding of chemical reactions, and, 
finally, for heating the given circuit section. 

The ratio of the power A P developed by a current in the volume 
&V of a conductor to the magnitude of this volume is called the 
unit power of the current P u corresponding to the given point of the 
conductor. By definition, the unit -power is 

= (5.34) 

Speaking conditionally, the unit power is the power developed in 
unit volume of a conductor. 

An expression for the unit power can be obtained proceeding from 
the following considerations. The force e (E + E*) develops a pow- 
er of 


P f = e (E + E*) (v + u) 

upon the motion of a current carrier. Let us average this expression 
for the carriers confined in the volume A V within which E and E* 
may be considered constant. The result is 

< P ' > = e"(E + E*) (v + u> = e (E-f E*) (v> + e (E + E*) (u> = 

= e (E + E*) {11 > 


(remember that (v) — 0). 

We can find the power A P developed in the volume AF by multi- 
plying ( P '} by the number of current carriers in this volume, i.e. by 
nAV (» is the number of carriers in unit volume). Thus, 

A P = </>•> nAF = e (E+E*) (u)nAF = j (E + E*) AF 
(see Eq. (5.23)1. Hence, 


P u = j(E + E*) (5.35) 


This expression is a differential form of the integral equation (5.33). 


112 


Electricity and Magnetism 


5.8. The Joule-Lenz Law 

When a conductor is stationary and no chemical transformations 
occur in it, the work of a current given by Eq. (5.32) goes to increase 
the internal energy of the conductor, and as a result the latter gets 
heated. It is customary to say that when a current flows in a conduc- 
tor, the heat 

Q = Ult 

is liberated. Substituting RI for U in accordance with Ohm’s law, 
we get the formula 

Q = RPt (5.36) 

Equation (5.36) was established experimentally by the British 
physicist James Joule (1818-1889) and independently of him by the 
Russian physicist Emil Lenz (1804-1865), and is called the Joule* 
Lenz law. 

If the current varies with time, then the amount of heat liberated 
during the time t is calculated by the equation 

t 

Q=^RI 2 dt (5.37) 

0 

We can pass over from Eq. (5.36) determining the heat liberated 
in an entire conductor to an expression characterizing the liberation 
of heat at different spots of the conductor. Let us separate in a con- 
ductor, in the same way as we did in deriving Eq. (5.22), an elemen- 
tary volume in the form of a cylinder (see Fig. 5.4). According to 
the Joule-Lenz law, the following amount of heat will be liberated 
in this volume during the time dt : 

dQ = RI 2 dt = (/ dS) 2 dt = p; 2 dV dt (5.38) 

(< dV = dS dl is the magnitude of the elementary volume). 

Dividing Eq. (5.38) by dV and dt , we shall find the amount of heat 
liberated in unit volume per unit time: 

Qu — PJ 2 (5.39) 

By analogy with the name of quantity (5.34), the quantity Q n 
can be called the unit thermal power of a current. 

Equation (5.39) is a differential form of the Joule-Lenz law. It can 
be obtained from Eq. (5.35). Substituting j la = pj for E + E* in 
Eq. (5.35) [see Eq. (5.25)], we arrive at the expression 

Pu — pi 2 

that coincides with Eq. (5.39). 

It must be noted that Joule and Lenz established their law for 
a homogeneous circuit section. As follows from what has been said 
in the present section, however, Eqs. (5.36) and (5.39) also hold for 
an inhomogeneous section provided that the extraneous forces acting 
in it have a non-chemical origin. 



CHAPTER 6 MAGNETIC FIELD 

IN A VACUUM 


6.1. Interaction of Currents 

Experiments show that electric currents exert a force on one anoth- 
er. For example, two thin straight parallel conductors carrying 
a current (we shall call them line currents) attract each other if the 
currents in them flow in the same direction, and repel each other if 
the currents flow in opposite directions. The force of interaction per 
unit length of each of the parallel conductors is proportional to the 
magnitudes of the currents I x and / 2 in them and inversely propor- 
tional to the distance b between them: 

F u = k — (6.1) 


We have designated the proportionality constant 2k for reasons that 
will become clear on a later page. 

The law of interaction of currents was established in 1820 by the 
French physicist Andre Ampere (1775-1836). A general expression 
of this law suitable for conductors of any shape will be given in 
Sec. 6.6. 

Equation (6.1) is used to establish the unit of current in the SI 
and in the absolute electromagnetic system (cgsm) of units. The 
SI unit of current — the ampere — is defined as the constant current 
which, if maintained in two straight parallel conductors of infinite 
length, of negligible cross section, and placed 1 metre apart in vacu- 
um, would produce between these conductors a force equal to 2 X 
X 10~ 7 newton per metre of length. 

The unit of charge, called the coulomb, is defined as the charge 
passing in 1 second through the cross section of a conductor in which 
a constant current of 1 ampere is flowing. Accordingly, the coulomb 
is also called the ampere-second (A*s). 

Equation (6.1) is written in the rational ize d form as follows: 


F u 


4 n b 


( 6 . 2 ) 


where \i 0 is the so-called magnetic constant [compare with Eq. (1.11)1. 
To find the numerical value of p 0 , we shall take advantage of the 
fact that according to the definition of the ampere, when I x = / 2 = 
= 1 A and b = 1 m. the force F u is obtained equal to 2 X 10“ 7 N/in. 



114 


Electricity and Magnetism 


Let us use these values in Eq. (6.2): 


2xiq-’^ 2 - x ; x - ! 

Hence, 

p 0 = 4n x 10" 7 = 1.26 x 10~ 6 H/m (6.3) 

(the symbol H/m stands for henry per metre — see Sec. 8.5). 

The constant k in Eq. (6.i) can be made equal to unity by choosing 
an appropriate unit of current. This is how the absolute electro- 
magnetic unit of current (cgsm x ) is established. It is defined as the 
current which, if maintained in a thin straight conductor of infinite 
length, would act on an equal and parallel line current at a distance 
of 1 cm from it with a force equal to 2 dyn per centimetre of length. 

In the cgse system/the constant k is a dimension quantity other 
than unity. According to Eq. (6.1), the dimension of k is determined 
as follows: 


[*] 


[*y>] 


in 

m* 


(6.4) 


We have taken into account that the dimension of F n is the dimen- 
sion of force divided by the dimension of length; hence, the dimen- 
sion of the product E u b is that of force. According to Eqs. (1.7) 
and (5.7): 

( F] = -gi; l/]=^-M. 

Using these values in Eq. (6.4), we find that 


[&] = 


x* 

L* 


Consequently, in the cgse system, k can be written in the form 


k = 


c* 


(6.5) 


where c is a quantity having the dimension of velocity and called 
the electromagnetic constant. To find its value, let us use rela- 
tion (1.8) between the coulomb and the cgse unit of charge, which was 
established experimentally. A force of 2 X 10~ 7 N/m is equivalent 
to 2 X 10“ 4 dyn/cm. According to Eq. (6.1), this is the force with 
which currents of 3 X 10® cgse x (i.e. 1 A) each interact when b = 
= 100 cm. Thus, 


whence 


1 2 X 3 X 10 ® X 3 X 10 ® 

ZX1U ~ c* Too 


c — 3 X 10 10 cm/s = 3 x 10 8 m/s (6.6) 


The value of the electromagnetic constant coincides with that of 
the speed of light in a vacuum. From J. Maxwell’s theory, there fol- 



Magnetic Field in a Vacuum 


115 


lows the existence of electromagnetic waves whose speed in a vacu- 
um equals the electromagnetic constant c. The coincidence of c 
with the speed of light in a vacuum gave Maxwell the grounds to 
assume that light is an electromagnetic wave. 

The value of k in Eq. (6.1) is 1 in the cgsm system and 1 lc 2 — 
= 1/(3 X 10 10 ) 2 sVcm 2 in the cgse system. Hence it follows that 


a current of i cgsmi is equivalent to a current of 3 X 

10 10 cgse x : 

1 cgsmj = 3 X 10 10 cgse x = 10 A 

(6.7) 

Multiplying this relation by 1 s, we get 


1 cgsm, = 3 X 10 10 cgse, = 10 C 

(6.8) 

Thus, 


r _ 1 _ j 

* cgsm — c 1 cgse 

(6.9) 

Accordingly, 


1 

<7cgsm — ~ ?cgse 

(6.10) 


There is a definite relation between the constants e 0 , p 0 , and c. 
To establish it, let us find the dimension and numerical value of the 
product e 0 p 0 . In accordance with Eq. (1.11), the dimension of e 0 is 

According to Eq. (6.2) 

< 6i2 > 

Multiplication of Eqs. (6.11) and (6.12) yields 

[eoM = -S-=- n gr (613) 

(v is the speed). 

With account taken of Eqs. (1.12) and (6.3), the numerical value 
of the product e 0 p 0 is 

e of*o = 4n x 9 x 10» x ^ nx 10" 7 = (3 x 10*)* "m 5 " (6.14) 

Finally, taking into account Eqs. (6.6), (6.13), and (6.14), we get 
the relation interesting us: 

eoHo=-^- (6.15) 


6.2. Magnetic Field 

Currents interact through a field called magnetic. This name origi- 
nated from the fact that, as the Danish physicist Hans Oersted 
(1777-1851) discovered in 1820, the field set up by a current has an 


116 


Electricity and Magnetism 


orienting action on a magnetic pointer. Oersted stretched a wire car- 
rying a current over a magnetic pointer rotating on a needle. When 
the cufrent was switched on, the pointer aligned itself at right angles 
to the wire. Reversing of the current caused the pointer to rotate in 
the opposite direction. 

Oersted’s experiment shows that a magnetic field has a sense of 
direction and must be characterized by a vector quantity. The latter 
is designated by the symbol B. It would be logical to call B the 
magnetic field strength, by analogy with the electric field strength E. 
For historical reasons, however, the basic force characteristic of 
a magnetic field was called the magnetic induction.The name magne- 
tic field strength was given to an auxiliary quantity H similar to 
the auxiliary characteristic D of an electric field. 

A magnetic field, unlike its electric counterpart, does not act on 
a charge at rest. A force appears only when a charge is moving. 

A current-carrying conductor is an electrically neutral system of 
charges in which the charges of one sign are moving in one direction, 
and the charges of the other sign in the opposite direction (or are at 
rest). It thus follows that a magnetic field is set up by moving 
charges. 

Thus, moving charges (currents) change the properties of the space 
surrounding them — they set up a magnetic field in it. This field 
manifests itself in that forces are exerted on charges moving in it 
(currents). 

Experiments show that the superposition principle holds for 
a magnetic field, the same as for an electric field: the field B set up 
by several moving charges {currents) equals the vector sum of the fields B, 
set up by each charge {current) separately : 

B = SB, (6.16) 

compare with Eq. (1.19)1. 


6.3. Field of a Moving Charge 

Space is isotropic, consequently, if a charge is stationary, then 
all directions have equal rights. This underlies the fact that the 
electrostatic field set up by a point charge is spherically symmet- 
rical. 

If a charge travels with the velocity v, a preferred direction (that 
of the vector v) appears in space. We can therefore expect the magne- 
tic field produced by a moving charge to have axial symmetry. We 
must note that we have in mind free motion of a charge, i.e. motion 
with a constant velocity. For an acceleration to appear, the charge 
must experience the action of a field (electric or magnetic). This 
field by its very existence would violate the isotropy of space. 



Magnetic Field in a Vacuum 


117 


Let us consider the magnetic field set up at point P by the point 
charge q travelling with the constant velocity v (Fig. 6.1). The distur- 
bances of the field are transmitted from point to point with th* 
finite velocity c. For this reason, the induction B at point P at the 
moment of time t is determined not by the position of the charge at 
the same moment t , but by its position at an 
earlier moment of time t — r: 

B (P, t) = f{q, y, r(f — t)} 

Here P signifies the collection of the coordinates 
of point P determined in a stationary reference 
frame, and r(£ — t) is the position vector drawn 
to point P from the point where the charge was 
at the moment t — t. 

If the velocity of the charge is much smaller 
than c (v c), then the retardation time t will 
be negligibly small. In this case, we can consider that the value of 
B at the moment t is determined by the position of the charge at 
the same moment U If this condition is observed, then 

B ( P , t ) = / {q, v, r (*)> (6.47) 

[we remind our reader that v = const, therefore v (t — t) = v (t)]. 

The form of function (6.17) can be established only experimentally. 
But before giving the results of experiments, let us try to find the 
logical form of this relation. The simplest assumption is that the 
magnitude of the vector B is proportional to the charge q and the 
velocity v (when v_^_0, a magnetic field is absent). We have to 
“construct” the vector B we are interestecTIn from the scalar q and 
the two given vectors v and r. This can be done by vector multipli- 
cation of the given vectors and then by multiplying their product by 
the scalar. The result is the expression 

q fvrl (6.18) 

The magnitude of this expression grows with an increasing distance 
from the charge (with increasing r). It is improbable that the charac- 
teristic of a field will behave in this way — for the fields that we 
know (electrostatic, gravitational), the field does not grow with an 
increasing distance from the source, but, on the contrary, weakens, 
varying in proportion to 1/r 2 . Let us assume that the magnetic 
field of a moving charge behaves in the same way when r changes. 
We can obtain an inverse proportion to the square of r by dividing 
Eq. (6.18) by r 3 . The result is 


P 



r» 


(6.49) 


118 


Electricity and Magnetism 


Experiments show that when v c, the magnetic induction of the 
field of a moving charge is determined by the formula 

B = (6.20) 


where k ' is a proportionality constant. 

We must stress once more that the reasoning which led us to 
expression (6.19) must by no means be considered as the derivation 
of Eq. (6.20). This reasoning does not have conclusive force. Its 
aim is to help us understand and memorize Eq. (6.20). This equation 
itself can be obtained only experimentally. 

It can be seen from Eq. (6.20) that the vector B at every point P 
is directed at right angles to the plane passing through the direction 
of the vector v and point P, so that rotation in the direction of B 
forms a right-handed system with the direction of v (see the circle 
with the dot in Fig. 6.1). We must note that B is a pseudo vector. 

The value of the proportionality constant k' depends on our choice 
of the units of the quantities in Eq. (6.20). This equation is written 
in the rationalized form as follows: 


T> Po Q t vr l 
4ji r* 

This equation can be written in the form 

B _ Po <7 [verl 


( 6 . 21 ) 

( 6 . 22 ) 


[compare with Eq. (1.15)1. It must be noted that in similar equations 
when e 0 is in the denominator, p 0 is in the numerator, and vice versa. 

The SI unit of magnetic induction is called the tesla (T) in honour 
of the Croatian electrician and inventor Nikola Tesla (1856-1943). 

The units of the magnetic induction B are chosen in the cgse and 
cgsm systems so that the constant k' in Eq. (6.20) equals unity. 
Hence, the same relation holds between the units of B in these sys- 
tems as between the units of charge: 

1 cgsm B = 3 x 10 i0 cgse B (6.23) 

[see Eq. (6.8)1. 

The cgsm unit of magnetic induction has a special name — the 
gauss (Gs). 

The German mathematician Karl Gauss (1777-1855) proposed a sys- 
tem of units in which all the electrical quantities (charge, current, 
electric field strength, etc.) are measured in cgse units, and all the 
magnetic quantities (magnetic induction, magnetic moment, etc.) 
in cgsm units. This system of units was named the Gaussian one, in 
honour of its author. 

In the Gaussian system, owing to Eqs. (6.9) and (6.10), all the 
equations containing the current or charge in addition to magnetic 


Magnetic Field in a Vacuum 


119 


quantities include one multiplier Me for each quantity I or q in the 
relevant equation. This multiplier converts the value of the perti- 
nent quantity (/ or q) expressed in cgse units to a value expressed in 
cgsm units (the cgsm system of units is constructed so that the pro- 
portionality constants in all the equations equal 1). For example, in 
the Gaussian system, Eq. (6.20) has the form 

B== T'^r^ ( 6 - 24 > 

We must note that the appearance of a preferred direction in space 
(the direction of the vector v) when a charge moves leads to the elec- 
tric field of the moving charge also losing its spherical symmetry and 



Fig. 6.2 


becoming axially symmetrical. The relevant calculations show that 
the E lines of the field of a freely moving charge have the form shown 
in Fig. 6.2. The vector E at point P is directed along the position 
vector r drawn from the point where the charge is at the given mo- 
ment to point P. The magnitude of the field strength is determined 
by the equation 


1 g 1 — v*/c* 

4jfe 0 [! — (i^/c*) sin 2 0J 3/2 


(6.25) 


where 0 is the angle between the direction of the velocity v and the 
position vector r. 

When y < c, the electric field of a freely moving charge at each 
moment of time does not virtually differ from the electrostatic 
field set up by a stationary charge at the point where the moving 
charge is at the given moment. It must be remembered, however, that 
this “electrostatic” field moves together with the charge. Hence, the 
field at each point of space changes with time. 

At values of v comparable with c, the field in directions at right 
angles to v is appreciably stronger than in the direction of motion at 



120 


Electricity and Magnetiem 


the same distance from the charge (see Fig. 6.2 drawn for vie — 0.8). 
The field “flattens out” in the direction of motion and is concentrated 
mainly near a plane passing through the charge and perpendicular 
to the vector v. 


6.4, The Biot-Savart Law 


Let us determine the nature of the magnetic field set up by an 
arbitrary thin wire through which a current flows. We shall consid- 
er a small element of the wire of length dl . 
This element contains nS dl current carriers 
(n is the number of carriers in a unit vol- 
JP ume, and S is the cross-sectional area of the 
wire where the element dl has been taken). 
At the point whose position relative to the 
element dl is determined by the position vec- 
tor r (Fig. 6.3), a separate carrier of current 
e sets up a field with the induction 



B = 


Ho e [(v + u), r] 
4 n r 3 


[see Eq. (6.21)]. Here v is the velocity of 
chaotic motion, and u is the velocity of 
« * ordered motion of the carrier. 

The value of the magnetic induction averaged over the current 
carriers in the element dl is 


(B) = 


Ho g[(W + W)t r l _ Ho *[(u)» r] 


4 n 


An 


((v> = 0). Multiplying this expression by the number of carriers 
in an element of the wire (equal to nS dl ), we get the contribution to 
the field introduced by the element dl: 

dB = (B) nS dl = 


(we have put the scalar multipliers n and e inside the sign of the 
vector product). Taking into account that ne(u> = j, we can write 


jg Ho & [j» r ] dl 
An r 3 


(6.26) 


Let us introduce the vector dl directed along the axis of the current 
element dl in the same direction as the current. The magnitude of 
this vector is dl. Since the directions of the vectors j and dl coincide, 
we can write the equation 


j dl = j d\ 


(6.27) 


Magnetic Field in a Vacuum 


121 


Performing such a substitution in Eq. (6.26), we get 


dB 


Ho Sj trfl, r( 
4ji r 3 


Finally, taking into account that the product Sj gives the current I 
in the wire, we arrive at the final expression determining the magnetic 
induction of the field set up by a current element of length dl: 

(6.28) 

We have derived Eq. (6.28) from Eq, (6.21). Equation (6.28) was 
actually established experimentally before Eq. (6.21) was known* 
Moreover, the latter equation was derived 
from Eq. (6.28). 

In 1820, the French physicists Jean Biot 
(1774-1862) and Felix Savart (1791-1841) 
studied the magnetic fields flowing along 
thin wires of various shape. The French 
astronomer and mathematician Pierre La- 
place (1749-1827) analysed the experimental 
data obtained and found that the magnetic 
field of any current can be calculated as the 
vector sum (superposition) of the fields set 
up by the separate elementary sections of 
the currents. Laplace obtained Eq. (6.28) 
for the magnetic induction of the field set 
up by a current element of length dl. In this 
connection, Eq. (6.28) is called the Biot- 
Savart- Laplace law, or more briefly the Biot-Savart law. 

A glance at Fig. 6.3 shows that the vector dB is directed at right 
angles to the plane passing through dl and the point for which the 
field is being calculated so that rotation about dl in the direction of 
dB is associated with dl by the right-hand screw rule. The magnitude 
of dB is determined by the expression 

= (6.29) 

where a is the angle between the vectors dl and r. 

Let us use Eq. (6.28) to calculate the field of a line current, i.e. the 
field set up by a current flowing through a thin straight wire of infi- 
nite length (Fig. 6.4). All the vectors dB at a given point have the 
same direction (in our case beyond the drawing). Therefore, addition 
of the vectors dB may be replaced with addition of their magnitudes. 
The point for which we are calculating the magnetic induction is at 
the distance b from the wire. 



122 


Electricity and Magnetism 


Inspection of Fig. 6.4 shows that 

6 , f rda 

r = — : , dl=— 

sin a ’ sin a 


b da 
sin 2 a 


Let us introduce these values into Eq. (6.29): 




Pq lb da sin a sin 2 a 
4 ji b 2 sin 2 a 


Mo 

4ji 


£ 

6 


sin a c?a 



The angle a varies within the limits from 0 to n for all the elements 
of an infinite line current. Hence, 

B = j dB = ltT I sin adct = -&-2- 
0 

Thus, the magnetic induction of the field of a line current is deter- 
mined by the formula 

S = (6-30) 

The magnetic induction lines of the field of a line current are a sys- 
tem of concentric circles surrounding the wire (Fig. 6.5). 


6.5. The Lorentz Force 

A charge moving in a magnetic field experiences a force which we 
shall call magnetic. The force is determined by the charge q , its 
velocity v, and the magnetic induction B at the point where the 
charge is at the moment of time being considered. The simplest 
assumption is that the magnitude of the force F is proportional to 
each of the three quantities < 7 , v, and B . In addition, F can be expect- 
ed to depend on the mutual orientation of the vectors v and B. 
The direction of the vector F should be determined by those of the 
vectors v and B. 

To “construct” the vector F from the scalar q and the vectors v 
and B, let us find the vector product of v and B and then multiply 



Magnetic Field in a Vacuum 


123 


the result obtained by the scalar q. The result is the expression 

q IvB] (6.31) 

It has been established experimentally that the force F acting on 
a charge moving in a magnetic field is determined by the formula 

F = kq [vB] (6.32) 

where A: is a proportionality constant depending on the choice of the 
units for the quantities in the formula. 

It must be borne in mind that the reasoning which led us to expres- 
sion (6.31) must by no means be considered as the derivation of 



Fig. 6.6 Fig. 6.7 


Eq. (6.32). This reasoning does not have conclusive force. Its aim 
is to help us memorize Eq. (6.32). The correctness of this equation 
can be established only experimentally. 

We must note that Eq. (6.32) can be considered as a definition of 
the magnetic induction B. 

The unit of magnetic induction B — the tesla — is determined so 
that the proportionality constant k in Eq. (6.32) equals unity. Hence, 
in SI units, this equation becomes 

F = q [vB] (6.33) 

The magnitude of the magnetic force is 

F = qvB sin a (6.34) 

where a is the angle between the vectors v and B. It can be seen 
from Eq. (6.34) that a charge moving along the lines of a magnetic 
field does not experience the action of a magnetic force. 

The magnetic force is directed at right angles to the plane contain- 
ing the vectors v and B. If the charge q is positive, then the direction 
of the force coincides with that of the vector [vB). When q is 
negative, the directions of the vectors F and [vBl are opposite 
(Fig. 6.6). 

Since the magnetic force is always directed at right angles to the 
velocity of a charged particle, it does no work on the particle. Hence, 
we cannot change the energy of a charged particle by acting on it 
with a constant magnetic field. 


124 


Electricity and Magnetism 


The force exerted on a charged particle that is simultaneously in 
an electric and a magnetic field is 


F == gE + q [vB] 


(6.35) 


This expression was obtained from the results of experiments by the 
Dutch physicist Hendrik Lorentz (1853-1928) and is called the 
Lorentz force. 

Assume that the charge q is moving with the velocity v parallel 
to a straight infinite wire along which the current I flows (Fig. 6.7). 

According to Eqs. (6.30) and (6.34), the 
charge in this case experiences a magnetic 
force whose magnitude is 

F = qvB = q v^*L (6.36) 

where b is the distance from the charge to 
the wire. The force is directed toward the 
wire when the charge is positive if the direc- 
tions of the current and motion of the charge 
are the same, and away from the wire if 
these directions are opposite (see Fig. 6.7). 
When the charge is negative, the direction of the force is reversed, 
the other conditions being equal. 

Let us consider two like point charges q x and q 2 moving along 
parallel straight lines with the same velocity v that is much smaller 
than c (Fig. 6 . 8 ). When Kc, the electric field does not virtually 
differ from the field of stationary charges (see Sec. 6.3). Therefore, 
the magnitude of the electric force F e exerted on the charges can be 
considered equal to 



Fig. 6.8 


F 


e. i 



(6.37) 


Equations (6.21) and (6.33) give us the following expression for 
the magnetic force F m exerted on the charges: 

F m . , = ^.2 = ^ = 5 - ^ (6.38) 


(the position vector r is perpendicular to v). 

Let us find the ratio between the magnetic and electric forces. It 
follows from Eqs. (6.37) and (6.38) that 

-77 = eoHoz ; 2 = 7 T (6-39) 

[see Eq. (6.15)1. We have obtained Eq. (6.39) on the assumption that 
v <C c. This ratio holds, however, with any i/s. 

The forces F e and F m are directed oppositely. Figure 6.8 has been 
drawn for like and positive charges. For like negative charges, the 



125 


Magnetic Field in a Vacuum 


directions of the forces will remain the same, while the directions of 
the vectors B x and B 2 will be reversed. For unlike charges, the direc- 
tions of the electric and magnetic forces will be the reverse of those 
shown in the figure. 

Inspection of Eq. (6.39) shows that the magnetic force is weaker 
than the Coulomb one by a factor equal to the square of the ratio of 
the speed of the charge to that of light. The explanation is that the 
magnetic interaction between moving charges is a relativistic effect 
(see Sec. 6.7). Magnetism would disappear if the speed of light were 
infinitely great. 


6.6. Ampere’s Law 

If a wire carrying a current is in a magnetic field, then each of 
the current carriers experiences the force 

F = el(y + u), B] (6.40) 

[see Eq. (6.33)1. Here v is the velocity of chaotic motion of a carrier, 
and u is the velocity of ordered motion. The action of this force is 
transferred from a current carrier to the conductor along which it is 
moving. As a result, a force acts on a wire with current in a mag- 
entic field. 

Let us find the value of the force dF exerted on an element of a wire 
of length dl. We shall average Eq. (6. 40) over the current carriers 
contained in the element dl: 

<F> = el((y) + (u>), B] = el (u>, Bl (6.41) 

(B is the magnetic induction at the place where the element dl is). 
The wire element contains riS dl carriers (n is the number of carriers 
in unit volume, and S is the cross-sectional area of the wire at the 
given place). Multiplying Eq. (6.41) by the number of carriers, we 
find the force we are interested in: 

dF = <F> nS dl = [{ne{ u>), Bl S dl 

Taking into account that ne<u) is the current density j, and S dl 
gives the volume of a wire element dV , we can write 

dF = [jBl dV (6.42) 

Hence, we can obtain an expression for the density of the force, i.e. for 
the force acting on unit volume of the conductor 

F u . v = [JBJ 

Let us write Eq. (6.42) in the form 

dF = [jBl S dl 


(6.43) 


126 Electricity and Magnetism 

Replacing in accordance with Eq. (6.27) jS dl with JS dl =* / dl, 
we arrive at the equation 

dF = I [dl, Bl (6.44) 

This equation determines the force exerted on a current element dl in 
a magnetic field. Equation (6.44) was established experimentally by 
Ampere and is called Ampere’s law. 

We have obtained Ampere’s law on the basis of Eq. (6.33) for the 
magnetic force. The expression for the magnetic force was actually 
obtained from the experimentally established equation (6.44). 



Fig. 6.9 Fig. 6.10 


The magnitude of the force (6.44) is calculated by the equation 

dF = IB dl sin a (6.45) 

where a is the angle between the vectors dl and B (Fig. 6.9). The force 
is normal to the plane containing the vectors dl and B. 

Let us use Ampere’s law to calculate the force of interaction be- 
tween two parallel infinitely long line currents in a vacuum. If the 
distance between the currents is b (Fig. 6.10), then each element of 
the current I 2 will be in a magnetic field whose induction is B x = 
= (|i 0 /4n) (2 IJb) [see Eq. (6.30)]. The angle a between the elements 
of the current / 2 and the vector B v is a right one. Hence, according to 
Eq. (6.45), the force acting on unit length of the current I 2 is 

F« (646) 

Equation (6.46) coincides with Eq. (6.2). 

We get a similar equation for the force Fit. u exerted on unit length 
of the current I v It is easy to see that when the currents flow in the 
same direction they attract each other, and in the opposite direction 
repel each other. 



Magnetic Field in a Vacuum 


127 


6.7. Magnetism as a Relativistic Effect 

There is a deep relation between electricity and magnetism. On the 
basis of the postulates of the theory of relativity and of the invari- 
ance of an electric charge, we can show that the magnetic interaction 
of charges and currents is a corollary of Coulomb’s law. We shall 


I 


B 



u 





a 


Fig. 6.11 


Fig. 6.12 


show this on the example of a charge moving parallel to an infinite 
line current with the velocity v 0 * (Fig. 6.11). 

According to Eq. (6.36), the magnetic force acting on a charge in 
the case being considered is 

F =* v * Ir-T < 6 ' 47 > 

(the meaning of the symbols is clear from Fig. 6.11). The force is 
directed toward the conductor carrying the current (q >0). 

Before commencing to derive Eq. (6.47) for the force on the basis 
of Coulomb’s law and relativistic relations, let us consider the 
following effect. Assume that we have an infinite linear train of 
point charges of an identical magnitude e spaced a very small distance 
l 0 apart (Fig. 6.12). Owing to the smallness of Z 0 , we can speak of the 
linear density of the charges X 0 which obviously is 

K=~ (6.48) 

*0 

Let us bring the charges into motion along the train with the identi- 
cal velocity u. The distance between the charges will therefore di- 
minish and become equal to 

Z = / 0 ]/ l--£ 

[see Eq. (8.19) of Vol. I, p. 231], The magnitude of the charges owing 
to their invariance, however, remains the same. As a result, the 
linear density of the charges observed in the reference frame relative 


* We have used the symbol for the velocity of a charge to make the nota- 
tion similar to that in Chap. 8 of Vol. I. 



128 


Electricity and Magnetism 


to which the charges are moving will change and become equal to 


^ = Xq 


(6.49) 


Now let us consider in the reference frame K two infinite trains 
formed by charges of the same magnitude, but of opposite signs, 
moving in opposite directions with the same velocity u and virtual- 


H 


+e 


© 

o 


e 

o 


-e 


© 

o 



b 


F= Wo B 



(a) 



K' 


*e 

© 

o 

-e 



* 

• o 



Fig. 6.13 


ly coinciding with each other (Fig. 6.13a). The combination of these 
trains is equivalent to an infinite line current having the value 

«w°> 

where % is the quantity determined by Eq. (6.49). The total linear 
density of the charges of a train equals zero, therefore an electric 
field is absent. The charge q experiences a magnetic force whose mag- 
nitude according to Eqs. (6.47) and (6.50) is 


r Po 

*n b /l — u*/c* 


4Xou 


(6.51) 


Let us pass over to the reference frame K r relative to which the 
charge q is at rest (Fig. 6.136). In this frame, the charge q also experi- 
ences a force (let us denote it by F'). This force cannot be of a magnet- 
ic origin, however, because the charge q is stationary. The force F 9 
has a purely electrical origin. It appears because the linear densities 
of the positive and negative charges in the trains are now different 
(we shall see below that the density of the negative charges is great- 
er). The surplus negative charge distributed over a train sets up 
an electric field that acts on the positive charge q with the force V 9 
directed toward the train (see Fig. 6.136). 

Let us calculate the force F' and convince ourselves that it “equals” 
— ***** v Hpt«rminftH hv Ect. f6.51L We have taken the word 


Magnetic Field in a Vacuum 


129 


•‘equals” in quotation marks because force is not an invariant quanti- 
ty. Upon transition from one inertial reference frame to another, the 
force transforms according to a quite complicated law. In a particular 
case, when the force F' is perpendicular to the relative velocity of the 
frames K and K' (F' _L v 0 ), the transformation has the form 

^ F' - + (F'v')/c* 

l + v 0 v7c* 


(v' is the velocity of a particle experiencing the force F' and mea- 
sured in the frame K‘). If v' =0 (which occurs in the problem we are 
considering), the formula for transformation of the force is as fol- 
lows: 

F = F']/ 1-£ 

A glance at this formula shows that the force perpendicular to v 0 
exerted on a particle at rest in the frame K ' is also perpendicular to 
the vector v 0 in the frame K . The magnitude of the force in this case, 
however, is transformed by the formula 

F = F' j/" 1--JJ (6.52) 

The densities of the charges in the positive and negative trains 
measured in the frame K' have the values [see Eq. (6.49)1 






^0 

Y l — u'Jfc* 


(6.53) 


where u\ and u'_ are the velocities of the charges + e and — e mea- 
sured in the frame K‘ . Upon a transition from the frame K to the 
frame A', the projection of the velocity of a particle onto the direction x 
coinciding with the direction of v 0 is transformed by the equation 


Uy. = 


u x — v o 
1 — UxVjc* 


[see Eqs. (8.28) of Vol. I, p. 237; we have substituted u and u* for v 
and v'\. For the charges the component u x equals u, for the 

charges — e it equals — u (see Fig. 6.13a). Hence, 


/ j7 m u u El. 

(Ux)+- ! — iiv o/ c* * V w x/- ““ i + ul , 0 / c i 

Since the remaining projections equal zero, we get 

, _ 1 U — Up 1 , __ U+Vp 

* 1 — uvjc* f - i-\-uupfc % 


(6.54) 



130 


Electricity and Magnetism 


To simplify our calculations, let us pass over to relative veloci- 
ties: 

o VO a “ Q' o' 

Po— c ' C » ~~~C~ * P- “ 

Equations (6.53) and (6.54) therefore acquire the form 

Xq ^ ' Xq 


K 


YT=v>' 

_ I P~Po I 


x -=-yr=m (655) 

^ :z WcT ’ = 1 + Ph (6 ' 56) 

With account taken of these equations, we get the following expres- 
sion for the total density of the charges: 

Xp x 0 


v = x;+v : 


l/l-f-P-Po -) 2 l/i-l P±Pjl\ 
v \ 1 — PPo ) V \ l + PP. ) 
_ Mi-PP,) 


(1 + PPo) 


Y( 1 - PPo) a ~ (P ~ Po) a / ( 1 + PPo) a - (P + Po) a 

It is easy to see that 

(1 - PPo)* - (P - p 0 ) 2 = (1 + PPo) 2 - (P + Po) 2 = (1 — Po) (1 — P 2 ) 

Consequently, 

___ — 2XqUVq 


Y( i— P8) (i — P 2 ) c*V i - "I/* 2 Y i — uVc* 


(6.57) 


In accordance with Eq. (1.122), an infinitely long filament carry- 
ing a charge of density X' sets up a field whose strength at the dis- 
tance b from the filament is 


£' = 


1 X' 


2 ne 0 b 

In this field, the charge q experiences the force 

qV 


F f = qE f = 


2ne 0 b 


Introduction of Eq. (6.57) yields (we have omitted the minus sign) 




q^QUV 0 


ne 0 6c 2 Y 1 — tfj/c 2 / 1 — u 2 /c* 

Po 


4Xoii 


(6.58) 


4n 6/1 — tt*/c 2 /l — i;8/c* 

I we remind our reader that p 0 = l/e 0 c 2 ; see Eq. (6.15)]. 

The expression obtained differs from Eq. (6.51) only in the factor 
1//1 — t#c*. We can therefore write that 

f=f' / 1-4 


where F is the force determined by Eq. (6.51), and F' is the force 
determined by Eq. (6.58). A comparison with Eq. (6.52) shows that F 



Magnetic Field in a Vacuum 


131 


and F* are the values of the same force determined in the frames K 
and K'. 

We must note that in the frame K ” which would move relative to 
the frame K with a velocity differing from that of the charge v 0 , the 
force exerted on the charge would consist of both electric and mag- 
netic forces. 

The results we have obtained signify that an electric and a magnetic 
field are inseparably linked with each other and form a single elec- 
tromagnetic field. Upon a special choice of the reference frame, a field 
may be either purely electric or purely magnetic. Relative to other 
reference frames, however, the same field is a combination of an. 
electric and a magnetic field. 

In different inertial reference frames, the electric and magnetic 
fields of the same collection of charges are different. A derivation 
beyond the scope of a general course in physics leads to the following 
equations for the transformation of fields when passing over from 
a reference frame if to a reference frame K' moving relative to it 
with the velocity v 0 : 


E- = E, 


B' 


Z?. E y — ”o B z jp. E z + v<> B y 

E *-YT=p' Ez ~ YT=P 

jy. By -f VqE z B z V 0 E U 

v ~ yt=p ’ YT=W J 


(6.59) 


Here E x , E yi E z , B x , B y , B 2 are the components of the vectors E 
and B characterizing an electromagnetic field in the frame K, similar 
primed symbols are the components of the vectors E' and B' charac- 
terizing the field in the frame K’. The Greek letter p stands for the 
ratio vjc. 

Resolving the vectors E and B, and also E' and B', into their com- 
ponents parallel to the vector v 0 (and, consequently, to the axes x 
and x') and perpendicular to this vector (i.e. representing, for exam- 
ple, E in the form E =Ej|+ E_l, etc.), we can write Eqs. (6.59) in. 
the vector form: 


Ef| = E m , E'j. = 

B |l — B ii» B i = 


E x + lv 0 BJ 
B a -(l/c»)[v 0 EJ 


(6.60)> 


In the Gaussian system of units, Eqs. (6.60) have the form 

^ Ex + d/Olv.BJ 

E|| = E„, E x = y== 

n n . B A -(l/e)lToEJ 


(6.61) 



132 


Electricity and Magnetism 


When P < 1 (i.e. v 0 <C c), Eqs. (6.60) are simplified as follows: 
E ll = E li» E x = E x + [v 0 B x ] 

B il = B||» = b jl — ■ ^-I voEj.) 

Adding these equations in pairs, we get 


E' = Ef| + E X == E||-4-E x + l v oBx] = E + (v 0 B x ] "J 
B'-Bh + BI-Bh+Bx— 3 -[vcE a 1 = B— ^-[v 0 E x ] j (6 * 62) 


Since the vectors v 0 and Bjj are collinear, their vector product 
equals zero. Hence, [v 0 B] = [v 0 B |l + [v 0 BjJ = [v 0 B_l1. Similarly, 
Iv 0 E] = [v 0 EjJ. With this taken into account, Eqs. (6.62) can be 
given the form 

E' = E + [V(,B], B' = B— [v 0 Ej (6.63) 


Fields are transformed by means of these equations if the relative 
velocity of the reference frames v 0 is much smaller than the speed 
of light in a vacuum c (i; 0 <C c). 

Equations (6.63) acquire the following form in the Gaussian system 
of units: 

E' = E + J-lv 0 B], B' = B-1 [v 0 E] (6.64) 

In the example in the frame K considered at the beginning of this 
section in which the charge q travelled with the velocity v 0 parallel 
to a current-carrying wire, there was only the magnetic field Bjl 
perpendicular to v 0 ; the components B n , E x and E ( | equalled zero. 
According to Eqs. (6.60) in the frame K' in which the charge q is at 
rest (this frame travels relative to K with the velocity v 0 ), the com- 
ponent Bj^ equal to Bj^/]/ 1 — p 2 is observed and, in addition, the 
perpendicular component of the electric field Ej^ = [v 0 BjJ/]^l — p 2 . 
In the frame K , the charge experiences the force 

F - q [v 0 BjJ (6.65) 

Since the charge q is at rest in the frame K' f it experiences in this 
frame only the electric force 

f ' =?e 1 = tSf (6 - 66) 

A comparison of Eqs. (6.65) and (6.66) yields F = F 'j/" 1 — p 2 f 
which coincides with Eq. (6.52). 



Magnetic Field in a Vacuum 


133 


6.8. Current Loop in a Magnetic Field 

Let us see. how a loop carrying a current behaves in a magnetic 
field. We shall begin with a homogeneous field (B= const). Accord- 
ing to Eq. (6.44), a loop element d\ experiences the force 

dF = / Id 1, B] (6.67) 

The resultant of such forces is 

F = <§/ Idl, B] (6.68) 

Putting the constant quantities / and B outside the integral, we get 

F = / l((£dl), B] 

The integral ^>dl equals zero, therefore. F = 0. Thus, the resultant 

force exerted on a current loop in a homogeneous magnetic field equals 
zero. This holds for loops of any shape (including non-planar ones) 
with an arbitrary arrangement of the loop relative to the direction 
of the field. Only homogeneity of the field is essential for the result- 
ant force to equal zero. 

In the following, we shall limit ourselves to a consideration of plane 
loops. Let us calculate the resultant torque set up by the forces 
(6.67) applied to a loop. Since the sum of these forces equals zero in 
a homogeneous field, the resultant torque relative to any point will 
be the same. Indeed, the resultant torque relative to point O is 
determined by the expression 

T = j [r, dF] 

where r is the position vector drawn from point O to the point of 
application of the force dF. Let us take point O' displaced relative 
to O by the distance b. Hence, r = b+r', and accordingly r' — 
= r — b. Therefore, the resultant torque relative to point O' is 

r = jtr\ dF] = j l(r - b), dF] = 

= j fr, dF] - j [b, dF] = T - [b, j dF] = T 

^ jdF = 0 j. The torques calculated relative to two arbitrarily taken 

points O and O' were found to coincide. We thus conclude that the 
torque does not depend on the selection of the point relative to which 
it is taken (compare with a couple of forces). 

Let us consider an arbitrary plane current loop in a homogeneous 
magnetic field B. Assume that the loop is oriented so that a positive 
normal to the loop n is at right angles to the vector B (Fig. 6.14). 
A normal is called* positive if its direction is associated with that of 
the current in the loop by the right-hand screw rule. 


134 


Electricity and Magnetism 


Let us divide the area of the loop into narrow strips of width dy 
parallel to the direction of the vector B (see Fig. 6.14a; Fig. 6.146 is 
an enlarged view of one of these strips). The force dF x directed beyond 
the drawing is exerted on the loop element dl t enclosing the strip at 
the left. The magnitude of this force is dF x = IB dl t sin a x = 
= IB dy (see Fig. 6.146). The force dF 2 directed toward us is exerted 
on the loop element dl 2 enclosing the strip at the right. The magnitude 

of this force is dF 2 = IB dL sina,= 
--- IB dy. 

The result we have obtained signi- 
fies that the forces applied to opposite 
loop elements dl 2 and dl 2 form a 
couple whose torque is 

) dT = IBx dy = IB dS 

{cLS is the area of a strip). A glance at 
Fig. 6.14 shows that the vector dT is 
perpendicular to the vectors n and B 
and, consequently, can be written in 
the form 

dT = / [nB] dS 




Summation of this equation over all the 
strips yields the torque acting on the 
loop: 

T = j7 [nBl dS=/ [nB] J dS=/[nB] S 

(6.69) 

(the field is assumed to be homogeneous, therefore the product [nB] 
is the same for all the strips and can be put outside the integral). 
The quantity S in Eq. (6.69) is the area of the loop. 

Equation (6.69) can be written in the form 


Fig. 6.14 


T = [(ISn), B] 


(6.70) 


This equation is similar to Eq. (1.58) determining the torque exerted 
on an electric dipole in an electric field. The analogue of E in 
Eq. (6.70) is the vector B, and that of the electric dipole moment p 
is the expression ISn. This served as the grounds to call the quan- 
tity 

Pm = ISn (6.71) 

the magnetic dipole moment of a current loop. The direction of the 
vector Pm coincides with that of a positive normal to the loop. 
Using the notation of Eq. (6.71), we can write Eq. (6.70) as follows: 

T = [p m , B] (p m _L B) (6.72) 



Magnetic Field in a Vacuum 


135 


Now let us assume that the direction of the vector B coincides 
with that of a positive normal to the loop n and, therefore, with that 
of the vector p m too (Fig. 6.15). In this case, the forces exerted on 
different elements of the loop are in one plane — that of the loop. The 




Fig. 6.16 


force exerted on the loop element d\ is determined by Eq. (6.67). 
Let us calculate the resultant torque produced by such forces relative 
to point O in the plane of the loop: 

T = J = J [r, <*F] = /$[r, Wl, B]] 

(r is the position vector drawn from point O to the element dl). Let 
us transform the integrand by means of Eq. (1.35) of Vol. I, p. 32. 
The result is 

T = I {<| (rB) d\ — <£ B (r, dl)} 

The first integral equals zero because the vectors r and B are mutual- 
ly perpendicular. The scalar product inside the second integral is 

r dr = 4- d (r 2 ). The second integral can therefore be written in the 

form 

yB§d(r*) 

The total differential of the function r 2 is inside the integral. The 
sum of the increments of a function along a closed path is zero. 
Hence, the second addend in the expression for T is zero too. We 
have thus proved that the resultant torque T relative to any point O 
in the plane of the loop is zero. The resultant torque relative to all 
other points has the same value (see above). 

Thus, when the vectors p m and B have the same direction, the 
magnetic forces exerted on separate portions of a loop do not tend 
to turn the loop nor shift it from its position. They only tend to 



136 


Electricity and Magnetism 


stretch the loop in its plane. If the vectors p m and B have opposite 
directions, the magnetic forces tend to compress the loop. 

Assume that the directions of the vectors p m and B form an arbitra- 
ry angle a (Fig. 6.16). Let us resolve the magnetic induction B into 
two components: By parallel to the vector p m and B x perpendicular 
to it, and consider the action of each component separately. The 
component B g will set up forces stretching or compressing the loop. 
The component B ± whose magnitude is B sin a will lead to the ap- 
pearance of a torque that can be calculated by Eq. (6.72): 

T = [ Pm , B J 

Inspection of Fig. 6.16 shows that 

IPmt Bj_l = IPmt Hi 

Consequently, in the most general case, the torque exerted on a plane 
current loop in a homogeneous magnetic field is determined by the 
equation 

T = [p m , B] (6.73) 

The magnitude of the vector T is 

T = p m B sin a (6.74) 

To increase the angle a between the vectors p m and B by da, the 
following work must be done against the forces exerted on a loop in 
a magnetic field: 

dA = T da = p m B sin a da (6.75) 

Upon turning to its initial position, a loop can return the work spent 
for its rotation by doing it on some other body. Hence, the work (6.75) 
goes to increase the potential energy W VtTaec h which a current loop 
has in a magnetic field, by the magnitude 

dFF p§ m ech = PmB sin a da 

Integration yields 

W p, m ech = — PmB cos a + const 
Assuming that const = 0, we get the following expression: 

W Pt m ech = — PmB cos a = — p m B (6.76) 

[compare with Eq. (1.61)1. 

Parallel orientation of the vectors p ra and B corresponds to the 
minimum energy (6.76) and, consequently, to the position of stable 
equilibrium of a loop. 

The quantity expressed by Eq. (6.76) is not the total potential 
energy of a current loop, but only the part of it that is due to the 
existence of the torque (6.73). To stress this, we have provided the 
symbol of the potential energy expressed by Eq. (6.76) with the 



Magnetic Field in a Vacuum 


137 


subscript “mech”. Apart from W p , me cht the total potential energy of 
a loop includes other addends. 

Now let us consider a plane current loop in an inhomogeneous 
magnetic field. For simplicity, we shall first consider the loop to be 
circular. Assume that the field changes the fastest in the direction x 
coinciding with that of B where the centre of the loop is, and that the 
magnetic moment of the loop is oriented along B (Fig. 6.17a). 

Here B ^ const, and Eq. (6.68) does not have to be zero. The force 
dF exerted on a loop element is perpendicular to B, i.e. to the 
magnetic field line where it intersects d\. Therefore, the forces ap- 
plied to different loop elements form a symmetrical conical fan 




(c) 


(Fig. 6.17b). Their resultant F is directed toward a growth in B and 
therefore pulls the loop into the region with a stronger field. It is 
quite obvious that the greater the field changes (the greater is dB/dx), 
the smaller is the apex angle of the cone and the greater, other condi- 
tions being equal, is the resultant force F. If we reverse the direction 
of the current (now p m is antiparallel to B), the directions of all the 
forces d F and of their resultant F will be reversed (Fig. 6.17c). Hence, 
with such a mutual orientation of the vectors p m and B, the loop 
will be pushed out of the field. 

It is a simple matter to find a quantitative expression for the force 
F by using Eq. (6.76) for the energy of a loop in a magnetic field. 
If the orientation of the magnetic moment relative to the field re- 
mains constant (a = const), then W Vt m ech will depend only on x 
(through B ). Differentiating W Vt meC h with respect to x and changing 
the sign of the result, we get the projection of the force onto the 
ar-axis: 


F x = 


mech 

di 


= Pm 


dB 

dx 


cos a 


We assume that the field changes only slightly in the other directions. 
Hence, we may disregard the projections of the force onto the other 



138 


Electricity and Magnetism 


axes and assume that F = F x . Thus, 

aD 

F = p m —cosa (6.77) 

According to the equation we have obtained, the force exerted on 
a current loop in an inhomogeneous magnetic field depends on the 
orientation of the magnetic moment of the loop relative to the direc- 
tion of the field. If the vectors p m and B coincide in direction (a = 
= 0), then the force is positive, i.e. is directed toward a growth in B 
(dB/dx is assumed to be positive; otherwise the sign and the direction 
of the force will be reversed, but the force will pull the loop into the 
region of a strong field as before). If p m and B are antiparallel (a = 
= jt), the force is negative, i.e. directed toward diminishing of B. 
We have already obtained this result qualitatively with the aid of 
Fig. 6.17. 

It is quite evident that apart from the force (6.77), a current loop 
in an inhomogeneous magnetic field will also experience the 
torque (6.73). 


6.9. Magnetic Field of a Current Loop 

Let us consider the field set up by a current flowing in a thin 
wire having the shape of a circle of radius R (a ring current). We shall 
determine the magnetic induction at the centre of the ring current 



dl 




(Fig. 6.18). Every current element produces at the centre an induc- 
tion directed along a positive normal to the loop. Therefore, vector 
summation of the dB's consists in summation of their magnitudes. 
By Eq. (6.29), 


dB 


Po Idl 
4 n R 2 


(a = ji/ 2). Let us integrate this expression over the entire loop: 

o f j d M’O f £ il — 2 jr/? 2 (InR 2 ) 

B— J dB ~ 4S"7P" < 5 )d *- 4iT R » 4n 



Magnetic Field in a Vacuum 


139 


The expression in parentheses is the magnitude of the magnetic dipole 
xnoment p m [see Eq. (6.71)]. Hence, the magnetic induction at the 
centre of a ring current has the value 


d Po ^Pm 

4n i ?3 


(6.78) 


Inspection of Fig. 6.18 shows that the direction of the vector B 
coincides with that of a positive normal to the loop, i.e. with that of 
the vector p m . Therefore, Eq. (6.78) can be written in the vector 


form: 


d Po 2p m 
4 n R* 


(6.79) 


Now let us find B on the axis of the ring current at the distance of r 
from the centre of the loop (Fig. 6.19). The vectors dB are perpendic- 
cular to the planes passing through the relevant element d\ and the 
point where we are seeking the field. Hence, they form a symmetrical 
conical fan (Fig. 6.196). We can conclude from considerations of 
symmetry that the resultant vector B is directed along the axis of 
the loop. Each of the component vectors dB contributes dB { j equal 
in magnitude to dB sin p = dB {Rib) to the resultant vector. The 
angle a between d\ and b is a right one, hence 


dB„ = dB£ 


f^o I dl H fio IH dl 

Tn~b*~~b~ 4it 


Integrating over the entire loop and substituting Y R 2 + r * for 6, 
we obtain 





IR 

6 s 


2 j xR = 


Po 2 ( InR 2 ) Po 2 Pm /o ori\ 

4ji (i ? 2 + r 2 ) 3 / 2 4n (i? a +r 2 ) $ / a 

This equation determines the magnitude of the magnetic induction on 
the axis of a ring current. With a view to the vectors B and p m having 
the same direction, we can write Eq. (6.80) in the vector form: 


Po 2p m 
4 n (i? 2 + r 3 ) 3 / 2 


(6.81) 


This expression does not depend on the sign of r. Hence, at points on 
the axis symmetrical relative to the centre of the current, B has the 
same magnitude and direction. 

When r = 0, Eq. (6.81) transforms, as should be expected, into 
Eq. (6.79) for the magnetic induction at the centre of a ring current. 

For great distances from a loop, we may disregard R 2 in the de- 
nominator in comparison with r 2 . Equation (6.81) now becomes 


B= 4 “““?“ (along the current axis) (6.82) 


which is similar to Eq. (1.55) for the electric field strength along the 
axis of a dipole. 


140 


Electricity and Magnetism 


Calculations beyond the scope of the present book show that a mag- 
netic dipole moment p m can be ascribed to any system of currents or 

moving charges localized in a restricted por- 
tion of space (compare with the electric dipole 
moment of a system of charges). The magnetic 
field of such a system at distances that are 
great in comparison with its dimensions is de- 
termined through p m using the same equations 
as those used to determine the field of a system 
of charges at great distances through the elec- 
tric dipole moment (see Sec. 1.10). In partic- 
ular, the field of a plane loop of any shape at 
great distances from it is 

B = It -7T- /i+3cos*e (6.83) 

where r is the distance from the loop to the 
given point, and 0 is the angle between the 
Fig. 6.20 direction of the vector p m and the direction 

from the loop to the given point of the field 
[compare with Eq. (1.53)1. When 0=0, Eq. (6.83) gives the same 
value as Eq. (6.82) for the magnitude of the vector B. 

Figure 6.20 shows the magnetic field lines of a ring current. It 
shows only the lines in one of the planes passing through the current 
axis. A similar picture will be observed in any of these planes. 

It follows from everything said in the preceding and this sections 
that the magnetic dipole moment is a very important characteristic 
of a current loop. It determines both the field set up by a loop and 
the behaviour of the loop in an external magnetic field. 

6.10. Work Done When a Current Moves 
in a Magnetic Field 

Let us consider a current loop formed by stationary wires and 
a movable rod of length l sliding along them (Fig. 6.21). Let the 
loop be in an external magnetic field which we shall assume to be 
homogeneous and at right angles to the plane of the loop. With the 
directions of the current and field shown in the figure, the force F 
exerted on the rod will be directed to the right and will equal 

F = IBl 

When the rod moves to the right by dh, this force does the positive 
work 

dA = F dh = IBl dh = IB dS (6.84) 

where dS is the hatched area (see Fig. 6.21a). 




Magnetic Field in a Vacuum 


141 


Let us see how the magnetic induction flux O through the area of 
the loop will change when the rod moves. We shall agree, when cal- 
culating the flux through the area of a current loop, that the quantity 
o in the equation 

<D = J Bn dS 

is a positive normal, i.e. one that forms a right-handed system with 
the direction of the current in the loop (see Sec. 6.8). Hence in the 
case shown in Fig. 6.21a, the flux will be positive and equal to BS 



(S is the area of the loop). When the rod moves to the right, the area 
of the loop receives the positive increment dS . As a result, the flux 
also receives the positive increment dtf) = BdS. Equation (6.84) 
can therefore be written in the form 

dA = I dd> (6.85) 

When the field is directed toward us (Fig. 6.216), the force exerted on 
the rod is directed to the left. Therefore when the rod moves to the 
right through the distance dh, the magnetic force does the negative 
work 

dA = —IBl dh = —IB dS (6.86) 

In this case, the flux through the loop is — BS. When the area of the 
loop grows by dS , the flux receives the increment d<S> = — B dS . 
Hence, Eq. (6.86) can also be written in the form of Eq. (6.85). 

The quantity d<Din Eq. (6.85) can be interpreted as the flux through 
the area covered by the rod when it moves. We can say accord- 
ingly that the work done by the magnetic force on a portion of 
a current loop equals the product of the current and the magnitude 
of the magnetic flux through the surface covered by this portion 
during its motion. 

Equations (6.84) and (6.86) can be combined into a single vector 
expression. For this purpose, we shall compare the vector 1 having 
the direction of the current with the rod (Fig. 6.22). Regardless of 
the direction of the vector B (toward us or away from us), the force 



142 


Electricity and Magnetism 


exerted on the rod can be represented in the form 

F = / [IB] 

When the rod moves through the distance dh, the force does the work 

dA = F dh = I [IB] dh 

Let us perform a cyclic transposition of the multipliers in this triple 



Fig. 6.22 


scalar product [see Eq. (1.34) of Vol. I, p. 311. The result is 

dA = IB [dh, 11 (6.87) 


A glance at Fig. 6.22 shows that the vector product [dh, 1] equals 
in magnitude the area dS described by the rod during its motion and 

has the direction of a positive normal n. 
*/ Hence, 


dA = /Bn dS (6.88) 

In the case shown in Fig. 6.22a, we have 
Bn = B, and we arrive at Eq. (6.84). In the 
Fig. 6.23 case shown in Fig. 6.226, we have Bn = — /?, 

and we arrive at Eq. (6.86). 

The expression Bn dS determines the increment of the magnetic 
flux through the loop due to motion of the rod. Thus, Eq. (6.88) can 
be written in the form of (6.85). But Eq. (6.88) has an advantage 
over (6.85) because we “automatically” get the sign of dO from it 
and, consequently, the sign of dA too. 

Let us consider a rigid current loop of any shape in an arbitrary 
magnetic field. We shall find the work done upon an arbitrary infi- 
nitely small displacement of the loop. Assume that the loop element 
dl was displaced by dh (Fig. 6.23). The magnetic force does the 
following work on it: 


d \ i 


dAei = / [dl, B] dh 


( 6 . 89 ) 


Here B is the magnetic induction at the place where the loop ele- 
ment dl is. 



Magnetic Field in a Vacuum 


143 


Performing a cyclic transposition of the multipliers in Eq. (6.89), 
wo 

dA el = IB{dh y dl] (6.90) 

The magnitude of the vector product [dh, dl] equals the area of a par- 
allelogram constructed on the vectors dh and dl, i.e. the area dS 
described by the element dl during its motion. The direction of the 
vector product coincides with that of a positive normal to the area 
dS. Consequently, 

B [ dh , dl] = Bn dS = d<D c , (6.91) 

where d<J> e j is the increment of the magnetic flux through the loop 
due to the displacement of the loop element dl. 

With a view to Eq. (6.91), we can write Eq. (6.90) in the form 

di4 e i = /d(Dei (6.92) 

Summation of Eq. (6.92) over all the loop elements yields an expres- 
sion for the work of the magnetic forces upon an arbitrary infinitely 
small displacement of the loop: 

cL4= j dA el = j / da>ei = / j d<D e , = IdO (6.93) 

(d<I> is the total increment of the flux through the loop). 

To find the work done upon a finite arbitrary di splacement of a loop, 
let us integrate Eq. (6.93) over the entire loop: 

A t2 = j dA = / j d<J> = / (0> 2 — O,) (6.94) 

Here O* and 0 2 are the values of the magnetic flux through the 
loop in its initial and final positions. The work done by the magnetic 
forces on the loop thus equals the product of the current and the 
increment of the magnetic flux through the loop. 

In particular, when a plane loop rotates in a homogeneous field 
from a position in which the vectors p m and B are directed oppositely 
(in this position cJ) = — BS) to a position in which these vectors 
have the same direction (in this position <I> = BS ), the magnetic 
forces do the following work on the loop: 

A = I{BS — (—BS)} = 2 IBS 

The same result is obtained with the aid of Eq. (6.91) for the poten- 
tial energy of a loop in a magnetic field: 

A = W lnit -W bn = PmB- ( -p m B ) = 2 p m B = 21 SB 

(Pm = IS). 

We must note that the work expressed by Eq. (6.94) is done not 
at the expense of the energy of the external magnetic held, but at 
the expense of the source maintaining a constant current in the 


144 


Electricity and Magnetism 


loop. We shall show in Sec. 8.2 that when the magnetic flux through 
a loop changes, an induced e.m.f. g* — — (d<t>/dt) is set up in the 
loop. Hence, the source in addition to the work done to liberate the 
Joule heat must also do work against the induced e.m.f. determined 
by the expression 

A = j dA= — J til dt—{ ~Idt= j IdO^I^-Gi) 
that coincides with Eq. (6.94). 


6.11. Divergence and Curl 
of a Magnetic Field 

The absence of magnetic charges in nature* results in the fact that 
the lines of the vector B have neither a beginning nor an end. There- 
fore, in accordance with Eq. (1.77), the flux of the vector B through 
a closed surface must equal zero. Thus, for any magnetic field and 
an arbitrary closed surface, the condition 

= <£ B dS = 0 (6.95) 

8 

is observed. This equation expresses Gauss’s theorem for the vector B: 
the flux of the magnetic induction vector through any closed surface 
equals zero. 

Substituting a volume integral for the surface one in Eq. (6.95) 
in accordance with Eq. (1.108), we find that 

j VB dV = 0 

The condition which we have arrived at must be observed for any 
arbitrarily chosen volume V. This is possible only if the integrand at 
each point of the field is zero. Thus, a magnetic field has the property 
that its divergence is zero everywhere: 

VB = 0 (6.96) 

Let us now turn to the circulation of the vector B. By definition, 
the circulation equals the integral 

§ B d\ (6.97) 


• The British physicist Paul Dirac made the assumption that magnetic charges 
(called Dirac's monopoles) should exist in nature. Searches for these charges 
have meanwhile given no results and the question of the existence of Dirac's 



Magnetic Field In a Vacuum 


145 


It is the simplest to calculate this integral for the field of a line 
current. Assume that a closed loop is in a plane perpendicular 
to the current (Fig. 6.24; the current is perpendicular to the plane of 
the drawing and is directed beyond the drawing). At each point of 
the loop, the vector B is directed along a tangent to the circumference 
passing through this point. Let us substitute B dl B for B dl in the 
expression for the circulation ( dl B is the projection of a loop element 
onto the direction of the vector B). Inspection of the figure shows that 



dl B equals b da, where b is the distance from the wire carrying the 
current to dl, and da is the angle through which a radial straight line 
turns when it moves along the loop over the element dl. Thus, intro- 
ducing Eq. (6.30) for B, we get 

Bdl=*5df B = ^--^&da=-^-da (6.98) 

With a view to Eq. (6.98), we have 

§Bdl=«-§L£da (6.99) 

Upon circumvention of the loop enclosing the current, the radial 
straight line constantly turns in one direction, therefore da = 2 n. 

Matters are different if the current is not enclosed by the loop 
(Fig. 6.246). Here upon circumvention of the loop, the radial straight 
line first turns in one direction (segment 1-2), and then in the 

opposite one (2-1), owing to which da equals zero. With a view to 
this result, we can write that 

B dl — fi 0 / (6.i00) 

where / must be understood as the current enclosed by the loop. If 
the loop does not enclose the current, the circulation of the vector B 
is zero. 



146 


Electricity and Magnetism 


The sign of expression (6.100) depends on the direction of circum- 
vention of the loop (the angle a is measured in the same direction). 
If the direction of circumvention forms a right-handed system with 
the direction of the current, quantity (6.100) is positive, in the oppo- 
site case it is negative. The sign can be taken into consideration by 
assuming / to be an algebraic quantity. A current whose direction is 



Fig. 6.25 


/ 



Fig. 6.26 


associated with that of circumvention of a loop by the right-hand 
screw rule must be considered positive; a current of the opposite 
direction will be negative. 

Equation (6.100) will allow us to easily recall Eq. (6.30) for B 
of the field of a line current. Imagine a plane loop in the form of 
a circle of radius b (Fig. 6.25). At each point of this loop, the vector B 
has the same magnitude and is directed along a tangent to the circle. 
Hence, the circulation equals the product of B and the length of the 
circumference 2 ji 6, and Eq. (6.100) has the form 

B X 2nb = p 0 / 

Thus, B = ii 0 I/2nb [compare with Eq. (6.30)1. 

The case of a non-planar loop (Fig. 6.26) differs from that of a plane 
one considered above only in that upon motion along the loop the 
radial straight line not only turns about the wire, but also moves 
along it. All our reasoning which led us to Eq. (6.100) remains true 
if we understand da to be the angle through which the projection of 
the radial straight line onto a plane perpendicular to the current 
turns. The total angle of rotation of this projection is 2 jt if the loop 
encloses the current, and zero otherwise. We thus again arrive at 
Eq. (6.100). 

We have obtained Eq. (6.100) for a line current. We can show that 
it also holds for a current Sowing in a wire of an arbitrary shape, for 
example for a ring current. 



Magnetic Field in a Vacuum 


147 


Assume that a loop encloses several wires carrying currents. Owing 
to the superposition principle [see Eq. (6.16)1: 

§ B<fl =§(2 B *)< fl =2§ B *<fl 

k k 

Each of the integrals in this sum equals |i 0 / k . Hence, 

§Bdl = Mo2 7 * (6.101) 

h 

(remember that I h is an algebraic quantity). 

If currents flow in the entire space where a loop is, the algebraic 
sum of the currents enclosed by the loop can be represented in the 
form 

2/*= J jdS= j jndS (6.102) 

k 8 8 

The integral is taken over the arbitrary surface S enclosing the 
loop. The vector j is the curr ent dens ity at the point where area 
element dS is; n is a positive normal To this element (i.e. a normal 
forming a right-handed system with the direction of circumvention 
of the loop in calculating the circulation). 

Substituting Eq. (6.102) for the sum of the currents in Eq. (6.101), 
we obtain 

^ BcQ = p 0 j j^S 
s 

Transforming the left-hand side according to Stokes’s theorem, 
we arrive at the equation 

j (VB] dS = Ho j jdS 

This equation must be obeyed with an arbitrary choice of the sur- 
face S over which the integrals are taken. This is possible only if the 
integrands have identical values at every point. We thus arrive at 
the conclusion that thejmrl of the magnetic induction vector is 
proportional to the current density vector at the given point: 

[VB] = p 0 j (6.103) 

The proportionality constant in the SI system is p 0 . 

We must note that Eqs. (6.101) and (6.103) hold only for the 
field in a vacuum in the absence of time-varying electric fields. 

Thus, we have found the divergence and curl of a magnetic field 
in a vacuum. Let us compare the equations obtained with the simi- 
lar equations for an electrostatic field in a vacuum. According to 



148 


Electricity and Magnetism 


Eqs. (1.117), (1.112), (6.96), and (6.103): 

vE = -^-P IVE| = 0 

(the divergence of E equals (the curl of E equals zero) 
p divided by e*) 

VB«=0 [VBJ = Hoi 

(the divergence of B equals (the curl of B equals 
zero) multiplied by p*) 

A comparison of these equations shows that an electrostatic and 
a magnetic field are of an appreciably different nature. The curl of 
an electrostatic field equals zero; consequently, an electrostatic 
field is potential and can be characterized by the scalar potential q>. 
The curl of a magnetic field at points where there is a current differs 
from zero. Accordingly, the circulation of the vector B is proportional 
to the current enclosed by a loop. This is why we cannot ascribe to 
a magnetic field a scalar potential that would be related to B by 
an equation similar to Eq. (1.41). This potential would not be 
unique — upon each circumvention of the loop and return to the 
initial point it would receive an increment equal to p 0 7. A field 
whose curl differs from zero is called a vortex or a solenotdal one. 

Since the divergence of the vector B is zero everywhere, this 
vector can be represented as the curl of a function A: 

B = IvA] (6.104) 

the divergence of a curl always equals zero; see Eq. (1.106)1. The 
function A is called the vector potential. A treatment of the vector 
potential is beyond the scope of the present book. 

6.12. Field of a Solenoid and Toroid 

A solenoid is a wire wound in the form of a spiral onto a round 
cylindrical body. The magnetic field lines of a solenoid are arranged 
approximately as shown in Fig. 6.27. The direction of these lines 
inside the solenoid forms a right-handed system with the direction of 
the current in the turns. 

A real solenoid has a current component along its axis. In addi- 
tion, the lin ear d ensity of the current j\\ n (equal to the ratio of the 
current dl to an element of solenoid length dl) changes periodically 
along the solenoid. The average value of this density is 

</«,„>= (jr)= nI < 6 - 105 ) 

where n = number of solenoid turns per unit length 
/ = current in the solenoid. 



Magnetic Field in a Vacuum 


149 


In .the science of electromagnetism, a great part is played by an 
imaginary infinitely long solenoid having no axial current component 
and, in addition, having a constant linear current density / lin along 
its entire length. The reason for this is that the field of such a sole- 
noid is homogeneous and is bounded by the volume of the solenoid 
(similarly, the electric field of an infinite parallel-plate capacitor 
is homogeneous and is bounded by the volume of the capacitor). 



Fig. 6.27 



In accordance with what has been said above, let us imagine a sole- 
noid in the form of an infinite thin-walled cylinder around which 
flows a current of constant linear density 

Zita = til ( 6 . 106 ) 

Let us divide the cylinder into identical ring currents — “turns’*. 
Examination of Fig. 6.28 shows that each pair of turns arranged 
symmetrically relative to a plane perpendicular to the solenoid 
axis sets up a magnetic induction parallel to the axis at any point 
of this plane. Hence, the resultant of the field at any point inside 
and outside an infinite solenoid can only have a direction parallel 
to the axis. 

It can be seen from Fig. 6.27 that the directions of the field inside 
and outside a finite solenoid are opposite. The directions of the fields 
do not change when the length of a solenoid is increased, and in 
the limit, when l — oo, they remain opposite. In an infinite sole- 
noid, as in a finite one, the direction of the field inside the solenoid 
forms a right-handed system with the direction in which the current 
flows around the cylinder. 

It follows from the vector B and the axis being parallel that the 
field both inside and outside an infinite solenoid must be homogeneous. 
To prove this., let us take an imaginary rectangular loop 1 -2-3-4 
inside a solenoid (Fig. 6-29; 4-1 is along the axis of the solenoid). 


150 


Electricity and Magnetism 


Passing clockwise around the loop, we get the value (B 2 — B t ) a 
for the circulation of the vector B. The loop does not enclose the 
currents, therefore the circulation must be zero (see Eq. (6.101)1. 
Hence it follows that = B 2 . Arranging section 2-3 of the loop at 
any distance from the axis, we shall always find that the magnetic 
induction B 2 at this distance equals the induction B x on the solenoid 
axis. Thus, the homogeneity of the field inside the solenoid has been 
proved. 

Now let us turn to loop l'-2'-3'-4\ We have depicted the vectors 
Bj and B' by a dash line since, as we shall find out in the following, 
the field outside an infinite solenoid is zero. Meanwhile, all that 


z\ 

f' 


V 

V 

1ST 

\ 2 




1 * 1 

.1 L 

H 




1 

I 

{ 


1 ! 



r 


Fig. 6.29 Fig. 6.30 

we know is that the possible direction of the field outside the solenoid 
is opposite to that of the field inside it. Loop l'-2' -3'-4' does not 
enclose the currents; therefore, the circulation of the vector B' 
around this loop, equal to ( B\ — S') a, must be zero. It thus follows 
that = B'. The distances from the solenoid axis to sections l'-4' 
and 2' -3' were taken arbitrarily. Consequently, the value of B' 
at any distance from the axis will be the same outside the solenoid. 
Thus, the homogeneity of the field outside the solenoid has been 
proved too. 

The circulation around the loop shown in Fig. 6.30 is a (B -1- B') 
(for clockwise circumvention). This loop encloses a positive current 
of magnitude ; lin a. In accordance with Eq. (6.101), the following 
equation must be observed: * , Jt — i 

a(B + B') = po/im a ^ 

or after cancelling a and replacing /u n with nl (see Eq. (6.106)] 

B + 5' = Pon/ (6.107) 

This equation shows that the field both inside and outside an infinite 
solenoid is finite. 



Magnetic Field in a Vacuum 


151 


Let us take a plane at right angles to the solenoid axis (Fig. 6.31). 
Since the field lines B are closed, the magnetic fluxes through the 
inner part S of this plane and through its outer part S' must be 
the same. Since the fields are homogeneous and normal to the plane, 
each of the fluxes equals the product of the relevant value of the 
magnetic induction and the area penetrated by the flux. We thus 
get the expression 

BS = B'S ' 

The left-hand side of this equation is finite, the factor S' in the 
right-hand side is infinitely great. Hence, it follows that B f — 0. 

Thus, we have proved that the magnetic induction outside an 
infinitely long soleno id is ze ro. The field inside the solenoid is homo- 



Fig. 6.31 Fig. 6.32 


geneous. Assuming in Eq. (6.107) that B' = 0, we arrive at an 
equation for the magnetic induction inside a solenoid: 

B = [i 0 nl (6.108) 

The product nl is called the number of ampere-turns per metre. 
At n = 1000 turns per metre and a current of 1 A, the magnetic 
induction inside a solenoid is 4jt X 10~ 4 T = 4n Gs. 

The symmetrically arranged turns make an identical contribution 
to the magnetic induction on the axis of a solenoid [see Eq. (6.81)]. 
Therefore, at the end of a se mi-i nfinite solenoid, the magnetic induc- 
tion on its axis equals half the value given by Eq. (6.108): 

B = y \i 0 nl (6. 109) 

Practically, if the length of a solenoid is considerably greater 
than its diameter, Eq. (6.108) will hold for points in the central 
part of the solenoid, and Eq. (6.109) for points on its axis near its 
ends. 



152 


Electricity anti Magnetism 


A toroid is a wire wound onto a body having the shape of a torus 
(Fig. 6.32). Let us take a loop in the form of a circle of radius r 
whose centre coincides with that of a toroid. Owing to symmetry, 
the vector B at every point must he directed along a tangent to the 
loop. Hence, the circulation of B is 

^ B dl = B X 2nr 

(B is the magnetic induction at the points through which the loop 
passes). 

If a loop passes inside a toroid, it encloses the current 2n Rnl 
(R is the radius of the toroid, and n is the number of turns per unit 
of its length). In this case 

Bx2nr = \i 0 2nRnI 

whence 

B = \ionI-^ ( 6 . 110 ) 

A loop passing outside a toroid encloses no currents, hence we 
have B X 2 nr = 0 for it. Thus, the magnetic induction outside 
a toroid is zero. 

For a toroid whose radius R considerably exceeds the radius of 
a turn, the ratio R/r for all the points inside the toroid differs only 
slightly from unity, and instead of Eq. (6.110) we get an equation 
coinciding with Eq. (6.108) for an infinitely long solenoid. In this 
case, the field may be considered homogeneous in each of the toroid 
sections. The field is directed differently in different sections. We 
can therefore speak of the homogeneity of the field within the entire 
toroid only conditionally, bearing in mind the identical magnitude 
of B. 

A real toroid has a current component along its axis. This compo- 
nent sets up a field similar to that of a ring current in addition to 
the field given by Eq. (6.110). 



CHAPTER 7 MAGNETIC FIELD 

IN A SUBSTANCE 


7.1. Magnetization of a Magnetic 

We assumed in the preceding chapter that the conductors carrying 
a current are in a vacuum. If the conductors carrying a current are 
in a medium, the magnetic field changes. The explanation is that 
any substance is a magnetic, i.e. is capable of acquiring a magnetic 
moment under the action of a magnetic field (of becoming magne- 
tized). The magnetized substance sets up the magnetic field B' that 
is superposed onto the field B 0 produced by the currents. Both fields 
produce the resultant field 

8=80 + 8 ' (7.1) 

{compare with Eq. (2.8)1. 

The true (microscopic) field in a magnetic varies greatly within 
the limits of intermolecular distances. By B is meant the averaged 
(macroscopic) field (see Sec. 2.3). 

To explain the magnetization of bodies, Ampere assumed that ring 
currents (molecular currents) circulate in the molecules of a sub- 
stance. Every such current has a magnetic moment and sets up a mag- 
netic field in the surrounding space. In the absence of an external 
field, the molecular currents are oriented chaotically, owing to which 
the resultant field set up by them equals zero. The total magnetic 
moment of a body also equals zero because of the chaotic orientation 
of the magnetic moments of its separate molecules. The action of 
a field causes the magnetic moments of the molecules to acquire 
a predominating orientation in one direction, owing to which the 
magnetic becomes magnetized — its total magnetic moment becomes 
other than zero. The magnetic fields of individual molecular currents 
in this case no longer compensate one another, and the field B r 
appears. 

It is quite natural to characterize the magnetization of a magnetic 
by the magnetic moment of unit volume. This quantity is called 
the magnetization and is denoted by the symbol M. If a magnetic 
is magnetized inhomogeneously, the magnetization at a given point 
is determined by the following expression: 

P® 


AF 


(7.2) 



154 


Electricity and Magnetism 


where A V is an infinitely small volume (from the physical viewpoint) 
taken in the vicinity of the point being considered, and p m is the 
magnetic moment of a separate molecule. Summation is performed 
over all the molecules confined in the volume AV [compare with 
Eq. (2.4)]. 

The field B' , like the field B 0 , h as no sour ces. Therefore, the diver- 
gence of the resultant field given by Eq. (7.1) is zero: 

VB = VB 0 + VB' = 0 (7 .3) 

Thus, Eq. (6.96) and, consequently, Eq. (6.95), hold not only for 
a field in a vacuum, but also for a field in a substance. 

7.2. Magnetic Field Strength 

Let us write an expression for the curl of the resultant field (7.1): 
[VB] = [ VB 0 ] + [ VB'] 

According to Eq. (6.103), IvB 0 ] = p 0 j, where j is the de nsity of 
the macroscopic current. Similarly, the curl of the vector B' must 
be proportional to the density of the mol ecul ar currents: 

[^B ] = H'ojmol 

Consequently, the curl of the resultant field is determined by the 
equation 

[VBj^Mi + imoi) (7.4) 

Inspection of Eq. (7.4) shows that when calculating the curl of 
a field in a magnetic, we encounter a difficulty similar to that which 
we encountered when dealing with an electric field in a dielectric 
[see Eq. (2.16)1: to determine the curl of B, we must know the den- 
sity not only of the macroscopic, but also of the molecular currents. 
But the density of the molecular currents, in turn, depends on the 
value of the vector B. The way of circumventing this difficulty is 
also similar to the one we took advantage of in Sec. 2.5. We are able 
to find such an auxiliary quantity whose curl is determined only 
by the density of the macroscopic currents. 

To find the form of this auxiliary quantity, let us attempt to 
express the density of the molecular currents j mo i through the mag- 
netization of a magnetic M (in Sec. 2.5 we expressed the density 
of the bound charges through the polarization of a dielectric P). 
For this purpose, let us calculate the algebraic sum of the molecular 
currents enclosed by a loop T. This sum is 

| jmol ^ (7.5) 

where S is the surface enclosing the loop. 



Magnetic Field in a Substance 


155 


The algebraic sum of the molecular currents includes only the 
molecular currents that are “threaded” onto the loop (see the current 
/moi * n Fig. 7.1). The currents that are not “threaded” onto the loop 
either do not intersect the surface enclosing the loop at all, or inter- 
sect it twice — once in one direction and once in the opposite one (see 
the current I~ m0 \ in Fig* 7.1). As a result, their contribution to the 
algebraic sum of the currents enclosed by the loop equals zero. 

A glance at Fig. 7.2 shows that the contour element dl making 
the angle a with the direction of magnetization M threads onto itself 


!e^ 




Fig. 7.1 Fig. 7.2 

those molecular currents whose centres are inside an oblique cylin- 
der of volume S moJ cos a dl (where £ mol is the area enclosed by a sep- 
arate molecular current). If n is the number of molecules in unit 
volume, then the total current enclosed by the element dl is 
/ m0 ]rtSmoi cos a d/. The product I mo \S mo i equals the magnetic mo- 
ment p m of an individual molecular current. Hence, the expression 
I m0 [S m0 \ n is the magnetic moment of unit volume, i.e. it gives 
the magnitude of the vector M, while Imo\S mo \n cos a gives the 
projection of the vector M onto the direction of the element dl. 
Thus, the total molecular current enclosed by the element dl is 
M dl, while the sum of the molecular currents enclosed by the entire 
loop (see Eq. (7.5)1 is 



Transforming the right-hand side according to Stokes’s theorem, 
we get 

f imoirfS= j [VM] dS 

S 8 

The equation which we have arrived at must be obeyed when the 
surface S has been chosen arbitrarily. This is possible only if the 
integrands are equal at every point of a magnetic: 

Jmnl = lVMl 


(7.6) 



156 


Electricity and Magnetism 


Thus, the density of the molecular currents is determined by the 
value of the curl of the magnetization. When [vM] = 0, the molec- 
ular currents of individual molecules are oriented so that their 
sum on an average is zero. 

Equation (7.6) allows us to make the following illustrative inter- 
pretation. Figure 7.3 shows the magnetization vectors M x and M 4 
in direct proximity to a certain point P. This point and both vectors 

are in the plane of the drawing. Loop T 
depicted by a dash line is also in the plane 
of the drawing. If the nature of the mag- 
netization is such that the vectors M x and 
M 2 are identical in magnitude, then the 
circulation of M around loop T will be 
zero. Accordingly, (vMl at point P will 
also be zero. 

The molecular currents i[ and i' 2 flow- 
ing in the loops depicted in Fig. 7.3 by 
solid lines can be compared with the mag- 
netizations M x and M 2 . These loops are in 
a plane normal to the plane of the draw- 
ing. With an identical direction of the 
vectors M x and M 2 , the directions of the currents i[ and i' at point P 
will be opposite. Since M x = M 2y the currents i\ and i\ are identical 
in magnitude, owing to which the resultant molecular current at 
point P, like IvM], will be zero: j mo i = 0. 

Now let us assume that Af x > M 2 . Therefore, the circulation of M 
around loop T will differ from zero. Accordingly, the field of the 
vector M at point P will be characterized by the vector [VM] directed 
beyond the drawing. A greater molecular current corresponds to 
a greater magnetization; hence, i[ >i v Consequently, at point P 
there will be observed a resultant current other than zero character- 
ized by the density j mol . The latter, like IvM), is directed beyond 
the drawing. When Af x < M 2y the vectors IvM] and j mol will be 
directed toward us instead of beyond the drawing. 

Thus, at points where the curl of the magnetization is other than 
zero, the density of the molecular currents also differs from zero, 
the vectors IvM) and j roo i having the same direction [see Eq. (7.6)]. 

Let us introduce Eq. (7.6) for the density of the molecular currents 
into Eq. (7.4): 

[VB] = p 0 j + Ho (VM] 

Dividing this equation by p* and combining the curls, we get 

[V. (£-«)]-. 


[vM] 



Fig. 7.3 


(7.7) 



Magnetic Field in a Substance 


157 


Whence it follows that 

H = i-M (7.8) 

is our required auxiliary quantity whose curl is determined only 
by the macroscopic currents. This quantity is called the magnetic 
field strength. 

In accordance with Eq. (7.7), 

[VH1 - j (7.9) 

(the curl of the vector H equals the vector of the density of the macro- 
scopic currents). 

Let us take an arbitrary loop T enclosed by surface S and form 
the expression 


j [VHjdS-J idS 

According to Stokes’s theorem, the left-hand side of this equation 
is equivalent to the circulation of the vector H around loop I\ 
Hence, 


$HdI 

r 



jdS 


(7.10) 


If macroscopic currents flow through wires enclosed by a loop, 
Eq. (7.10) can be written in the form 

$Hdl=2J* (7.11) 

h 

Equations (7.10) and (7.11) express the theorem on the circulation 
of the vector H: the circulation of the magnetic field strength vector 
around a loop equals the algebraic sum of the macroscopic currents 
enclosed by this loop. 

The magnetic field strength His the analogue of the electric displace- 
ment D. It was] originally assumed that magnetic masses similar 
to electric charges exist in nature, and the science of magnetism 
developed along the lines of that of electricity. Back in those times, 
the relevant names were introduced: the “magnetic induction” for 
B and the ‘‘field strength” (formerly “field intensity”) for H. It was 
later established that no magnetic masses exist in nature and that 
the quantity called the magnetic induction is actually the analogue 
not of the electric displacement D, but of the electric field strength E 
(accordingly, H is the analogue of D instead of E). It was decided 
not to change the established terminology, however, moreover 
because owing to the different nature of an electric and a magnetic 


158 


Electricity and Magnetism 


field (an electrostatic field is potential, a magnetic one is solenoidal*)^ 
the quantities B and D display many similarities in their behaviour 
(for example, the B lines, like the D lines, are not disrupted at the 
interface between two media). 

In a vacuum, M — 0, therefore H transforms into B/|x 0 and Eqs. 
(7.9) and (7.11) transform into Eqs. (6.103) and (6.101). 

In accordance with Eq. (6.30), the strength of the field of a line 
current in a vacuum is determined by the expression 




An 


2 / 

b 


(7.12) 


whence it can be seen that the magnetic field strength has a dimension 
equal to that of current divided by that of length. In this connection, 
the SI unit of magnetic field strength is called the ampere per me- 
tre (A/m). 

In the Gaussian system, the magnetic field strength is defined 
as the quantity 

H = B — 4jiM (7.13) 


It follows from this definition that in a vacuum H coincides with B. 
Accordingly, the unit of H in the Gaussian system, called the oersted 
(Oe), has the same value and dimension as the unit of magnetic 
induction— the gauss (Gs). In essence, the oersted and gauss are 
different names of the same unit. If the latter measures H, it is 
called the oersted, and if it measures B — the gauss. 

I It is customary practice, to associate the magnetization not with 
\ the magnetic induction, but with the field strength. It is assumed 
' that at every point of a magnetic 


M = XmH (7- 14) 

where X m is a quantity characteristic of a given magnetic and called 
the magnetic susceptibility**. Experiments show that for weakly 
magnetic (non-ferromagnetic) substances in not too strong fields 
Xm is independent of H. According to Eq. (7.8), the dimension of H 
coincides with that of M. Hence, % m is a dimensionless quantity. 
Using Eq. (7.14) for M in Eq. (7.8), we get 


whence 




H 


B 

Po (1 + Xm) 


(7.15) 


* A solenoidal field is one having no sources. At each point of such a field, 
the divergence is zero. 

♦* In anisotropic media, the directions of the vectors M and H, generally 
speaking, do not coincide. For such media, the relation between the vectors 
M and H is achieved by means of the magnetic susceptibility tensor (see the foot- 
note on p. 63). 



Magnetic Field in a Substance 


159 


The dimensionless quantity 

P = 1 + Xm 


(7.16) 


is called the relative permeability or simply the permeability of a sub- 
stance*. 

Unlike the dielectric susceptibility x that can have only positive 
values (the polarization P in an isotropic dielectric is always direct- 
ed along the E field), the magnetic susceptibility x m may be either 
positive or negative. Hence, the permeability may be either greater 
or smaller than unity. 

With account taken of Eq. (7.16), Eq. (7.15) can be written as 
follows: 


H = 


B 

Mol* 


(7.17) 


Thus, the magnetic field strength H is a vector having the same 
direction as the vector B, but whose magnitude is p 0 p times smaller 
(in anisotropic media the vectors H and B, generally speaking, do 
not coincide in direction). 

Equation (7.14) relating the vectors M and H has exactly the 
same form in the Gaussian system too. Using this equation in 
Eq. (7.13), we get 


H = B - 4n Xm H 

whence 

TT B 

l + 4»Xm 


(7.18) 


The dimensionless quantity 


p = 1 + 4nx m (7.19) 

is called the permeability of a substance. Introducing this quantity 
into Eq. (7.18), we get 

H = f- (7.20) 

The value of p in the Gaussian system of units coincides with 
its value in the SI. A comparison of Eqs. (7.16) and (7.19) shows 
that the value of the magnetic susceptibility in the SI is 4n times 
that of Xm Gaussian system: 


Xm, SI ^^Xm, G> 


(7.21) 


• The so-called absolute permeability p a = p 0 p is introduced in electrical 
engineering. This quantity is deprived of a physical meaning, however, and 
we shall not use it. 


160 


Electricity and Magnetism 


7.3. Calculation of the Field 
in Magnetics 

Let us consider the field produced by an infinitely long round 
magnetized rod. We shall consider the magnetization M to be the 
same everywhere and directed along the axis of the rod. Let us 
divide the rod mentally into layers of thickness dl at right angles 




Fig. 7.4 

to the, axis. We shall divide each layer in turn into small cylindrical 
elements with bases of an arbitrary shape and of area dS (Fig. 7.4a). 
Each such element has the magnetic moment 

dp m =M dSdl (7.22) 

The field dB' set up by an element at distances that are great in 
comparison with its dimensions is equivalent to the field that would 
produce the current I = M dl flowing around the element along its 
side surface (see Fig. 7.46). Indeed, the magnetic moment of such 
a current is dp m = I dS = M dl dS (compare with Eq. (7.22)], 
while the magnetic field at great distances is determined only by the 
magnitude and direction of the magnetic moment (see Sec. 6.9). 

The imaginary currents flowing in the section of the surface com- 
mon for two adjacent elements are identical in magnitude and oppo- 
site in direction, therefore their sum is zero. Thus, when summating 
the currents flowing around the side surfaces of the elements of one 
layer, only the currents flowing along the side surface of the layer 
will remain uncompensated. 

It follows from the above that a rod layer of thickness dl sets up 
a field equivalent to the one which would be produced by the cur- 
rent M dl flowing around the layer along its side surface (the linear 
density of this current is jn n = M ). The entire infinite magnetized 
rod sets up a field equivalent to the field of a cylinder around which 
flows a current saving the linear density ; lln = M . We established 


Magnetic Pield tn a Substance 


161 


in Sec. 6.12 that outside such a cylinder the field vanihes, while 
in 9 ide it the held is homogeneous and equals (Jt 0 /n a in magnitude. 

We have thus determined the nature of the held B' set up by a homo- 
geneously magnetized infinitely long round rod. Outside the rod, 
the field vanishes. Inside it, the field is homogeneous and equals 

B' = p 0 M (7.23) 

Assume that we have a homogeneous field B„ set up by macrocur- 
rents in a vacuum. According to Eq. (7.17), the strength of this 
field is 

wi 

Let us introduce into this field (we shall call it an external one) an 
infinitely long round rod of a homogeneous and isotropic magnetic, 
arranging it along the direction of B 0 . It follows from considerations 
of symmetry that the magnetization M set up in the rod is collinear 
with the vector B 0 . 

The magnetized rod produces inside itself the field B' determined 
by Eq. (7.23). The field inside the rod, as a result, becomes equal to 

B = B 0 + B = Bo (LtoM (7.25) 

Using this value of B in Eq. (7.8), we get the strength of the field 
inside the rod 

H — — — M = -5s. = H 0 
Ho Ho 0 

{see Eq. (7.24)1. Thus, the strength of the field in the rod coincides 
with that of the external field. 

Multiplying H by |x 0 p. we get the magnetic induction inside the rod: 

B = pbpH = HoH - jjj- = H®o (7.26) 

Hence, it follows that the permeability jx shows how many times 
the field increases in a magnetic [compare with Eq. (2.33)1. 

It must be noted that since the field B' is other than zero only inside 
the rod, the magnetic field outside the rod remains unchanged. 

The result we have obtained is correct when a homogeneous and 
isotropic magnetic fills the volume bounded by surfaces formed 
by the strength lines of the external field*. Otherwise the field 
strength determined by Eq. (7.8) does not coincide with H 0 = B 0 /|x 0 . 

It is conditionally assumed that the field strength in a magnetic is 

H = H 0 — H d (7.27) 

* We remind our reader that for an electric field D = Dp provided that a 
homogeneous and isotropic dielectric fills the volume bounded by equipoten- 
tial surfaces, i.e. surfaces orthogonal to the strength lines of the external field* 



162 


Electricity and Magnetism 


where H 0 is the external field, and H d is the so-called demagnetizing 
field. The latter is assumed to be proportional to the magnetization 

H d = JVM (7.28) 

The proportionality constant N is known as the demagnetization 
factor. It depends on the shape of a magnetic. We have seen that 
H = H 0 for a body whose surface is not intersected by strength 
lines of the external field, i.e. the demagnetization factor is zero. 
For a thin disk perpendicular to the external field, N = 1, and for 
a sphere, N = 1/3. 

The relevant calculations show that when a homogeneous and 
isotropic magnetic having the shape of an ellipsoid is placed in a homo- 
geneous external field, the magnetic field in it is also homogeneous, 
although it differs from the external one. This also holds for a sphere, 
which is a particular case of an ellipsoid, and for a long rod and 
a thin disk, which can be considered as the extreme cases of an ellip- 
soid. 

In concluding, let us find the field strength of an infinitely long 
solenoid filled with a homogeneous and isotropic magnetic (or sub- 
merged in an infinite homogeneous and isotropic magnetic). Applying 
the theorem on circulation [see Eq. (7.11)1 to the loop shown in 
Fig. 6 .30, we get the equation Ha = nal . Hence, 

H = nl (7.29) 

Thus, the field strength inside an infinitely long solenoid equals 
the product of the current and the number of turns per unit length. 
Outside the solenoid, the field strength vanishes. 


7.4. Conditions at the Interface 
of Two Magnetics 

Near the interface of two magnetics, the vectors B and H must 
comply with definite boundary conditions that follow from the 
relations 

VB = 0, [VH1 = j (7.30) 

[see Eqs. (7.3) and (7.9)1. We are considering stationary fields, 
i.e. ones that do not vary with time. 

Let us take on the interface of two magnetics of permeabili- 
ties pj and p, an imaginary cylindrical surface of height h with 
bases and S t at different sides of the interface (Fig. 7.5). The 
flux of the vector B through this interface is 

= B if n S + B z , n S -J- {B n ) “^slde (7.31) 

[compare with Eq. (2.46)1. 



Magnetic Field in a Substance 


163 


Since VB *=■ 0, the flux of the vector B through any closed surface 
is zero. Equating expression (7.31) to zero and making the transition 
we arrive at the equation B x%n = — If we project 
and B 2 onto the same normal, we get the condition 

*,.n = *2.n (7.32) 


1 compare with Eq. (2.47)]. 

Replacing in accordance with Eq. (7.17) the components of B 
with the corresponding components of H multiplied by p 0 p, we get 
the equation 


whence 


H l ,n _ M» 

Mi 


(7.33) 


Now let us take a rectangular loop on the interface of the magnetics 
(Fig. 7.6) and calculate the circulation of H for it. With small diraen- 




Fig. 7.6 


sions of the loop, the circulation can be written in the form 

^ H i dl = H i% x a — H 2t x a + (Hi) 2b (7.34) 

where (H t ) is the average value of H| on the parts of the loop at 
right angles to the interface. If no macroscopic currents flow along 
the interface of the magnetics, fvH] within the limits of the loop 
will equal zero. Consequently, the circulation will also be zero. As- 
suming that Eq. (7.34) is zero and performing the limit transition 
6 0, we arrive at the expression 

H„ = H 2tX (7.35) 


[compare with Eq. (2.44)]. 

Replacing the components of H with the corresponding components 
of B divided by p 0 p,, we get the relation 

^l.T _ T 

LLnLU 


LLaLL* 


164 


Electricity and Magnetism 


whence it follows that 


B 


l.T 


Hi 




(7.36) 


Summarizing, we can say that in passing through the interface 
between two magnetics, the normal component of the vector B 
and the tangential component of the vector H change continuously. 
The tangential component of the vector B and the normal component 
of the vector H in passing through the interface of the magnetics, 
„ however, experience a discontinuity. 

Thus, when passing through the inter- 
face of two media, the vector B behaves 
similar to the vector D, and the vector H 
similar to the vector E. 

Figure 7.7 shows the behaviour of the 
B lines when intersecting the surface be- 
tween two magnetics. Let the angles be- 
tween the B lines and a normal to the 
interface be and a 2 , respectively. The 
ratio of the tangents of these angles is 

tan ax B 1<T /Bi,n 

tan 04 Bj^f/Bj.n 

whence with a view to Eqs. (7.32) and (7.36) we get a law of refrac- 
tion of the magnetic field lines similar to Eq. (2.49): 



\L 

1 

l 

1 


1 



i \ 


Fig. 7.7 


tan a t p t 

tana 2 p 2 


(7.37) 


Upon passing into a magnetic with a greater value of p, the 
magnetic field lines deviate from a normal to the surface. This leads 
to crowding of the lines. The crowding of the B lines in a substance 
with a great permeability makes it possible to form magnetic beams, 
i.e. impart the required shape and direction to them. In particular, 
for magnetic shielding of a space, it is surrounded with an iron screen. 
A glance at Fig. 7.8 shows that the crowding of the magnetic field 
lines in the body of the screen results in weakening of the field in- 
side it. 

Figure 7.9 is a schematic view of a laboratory electromagnet. 
It consists of an iron core onto which coils supplied with a current 
are fitted. The magnetic field lines are mainly concentrated inside 
the core. Only in the narrow air gap do they pass in a medium with 
a low value of p. The vector B intersects the boundaries between the 
air gap and the core along a normal to the interface. It thus follows 
in accordance with Eq. (7.32) that the magnetic induction in the gap 
and in the core is identical in value. Let us apply the theorem on 
the circulation of H to the loop along the axis of the core. We can 



Magnetic Field in a Substance 


165 


assume that the field strength is identical everywhere in the iron and 
is /^tron ~ ^/MoMiron- In the air. Hair ~ ^^M-oM , air* Let us denote 
the length of the loop section in the iron by Z iron , and in the gap 
by fair- The circulation can thus be written in the form iron /iron + 




Fig; 7.9 


+ tfairfair- According to Eq. (7.11), this circulation must equal 
NI y where N is the total number of turns of the electromagnet coils, 
and / is the current. Thus, 


Hence, 


B 


MoMlron 




B = n 0 I 


TV 

ialr | hron 
Malr Miron 


fio/ 


TV 


ialrH* 


hron 

Miron 


(p a ir differs from unity only in the fifth digit after the decimal point). 

Usually, /air is of the order of 0.1 m, Z iron is of the order of 1 ra, 
while Miron reaches values of the order of several thousands. We 
may therefore disregard the second addend in the denominator and 
write that 

5=fio/ ^ (7 - 38> 

Consequently, the magnetic induction in the gap of an electromagnet 
has the same value as it would have inside a toroid without a core 
when A7/ a ir turns are wound on the torus per unit length [see Eq. 
(6.110)1. By increasing the total number of turns and reducing the 
dimensions of the air gap, we can obtain fields with a high value of 
B. In practice, fields with B of the order of several teslas (several 
tens of thousands of gausses) are obtained with the aid of electro- 
magnets having an iron core. 




166 


Electricity and Magnetism 


7.5. Kinds of Magnetics 

Equation (7.14) determines the magnetic susceptibility Xm of 
a unit volume of a substance. This susceptibility is often replaced 
with the molar (for chemically simple substances — the atomic) 
susceptibility Xm.moi’ (Xm, at) related to one* mole of a substance. 
It is evident that Xm.moi ~ Xm^moi* where V mo y is the volume of 
a mole of a substance. Whereas Xm is a dimensionless quantity, 
Xm, moi is measured in m 3 /mol. 

Depending on the sign and magnitude of the magnetic suscep- 
tibility, all magnetics are divided into three groups: 

(1) diamagnetics, for which Xm is negative and small in absolute 
value (Ixm, moi I is about 10 -11 to 10“ 10 m 3 /mol); 

(2) paramagnetics, for which Xm is also not great, but positive 
(Xm.moi is about 10- 10 to 10-* m 3 /mol); 

(3) ferromagnetics, for which Xm is positive and reaches very 
great values (Xm.moi is about 1 m 3 /mol). In addition, unlike dia- 
and paramagnetics for which Xm does not depend on H, the suscep- 
tibility of ferromagnetics is a function of the magnetic field strength. 

Thus, the magnetization M in isotropic substances may either 
•coincide in direction with H (in para- and ferromagnetics), or be 
directed oppositely to it (in diamagnetics). We remind our reader 
that in isotropic dielectrics the polarization is always directed in 
the same way as E. 

7.6. Gyromagnetic Phenomena 


The nature of molecular currents became clear after the British 


physicist Ernest Rutherford (1871-1937) established experimentally 
that the atoms of all substances consist of a positively charged 



Fig. 7.10 


nucleus and negatively charged electrons travel- 
ling around it. 

The motion of electrons in atoms obeys quantum 
laws; in particular, the concept of a trajectory can- 
not be applied to the electrons travelling in an 
atom. The diamagnetism of a substance can be ex- 
plained, however, by using the very simple Bohr 
model of an atom. According to this model, the 
electrons in atoms travel along stationary circular 


orbits. 


Assume that an electron is moving with the speed y in an orbit 
of radius r (Fig. 7.10). The charge ev, where e is the charge of an elec- 
tron and v is its number of revolutions a second, will be carried through 
an area at any place along the path of the electron in one second 



Magnetic Field in a Substance 


167 


Hence, an electron travelling in orbit will form the ring current 
/ = ev. Since the charge of an electron is negative, the direction 
of motion of the electron and the direction of the current will be oppo- 
site. The magnetic moment of the current set up by an electron is 

Prn = ts = e\nr 2 


The product 2nrv gives the speed of the electron i\ therefore we can 
write that 

i l-Z'" 

m <? 


00 


evr 

Pm — ~2” 


(7.39) 


The moment (7.39) is due to the motion of an electron in orbit and 
is therefore called the orbital magnetic moment. The direction 
of the vector p m forms a right-handed system with the direction of 
the current, and a left-handed one with th$t of motion of the elec- 
tron (see Fig. 7.10). 

An electron moving in orbit has the angular momentum 


L — mvr 


(7.40) 


(/ 7 i is the mass of an electron). The vector L is called the orbital 
angular momentum of an electron. It forms a right-handed system 
with the direction of motion of the electron. Hence, the vectors pm 
and L are directed oppositely. 

The ratio of the magnetic moment of an elementary particle to 
its angular momentum is called the gyromagnetic (or magnetomecha- 
nical) ratio. For an electron, it is 


Pm £_ 

L 2m 


(7.41) 


(m is the mass of an electron; the minus sign indicates that the mag- 
netic moment and the angular momentum are directed oppositely). 

Owing to its rotation about the nucleus, an electron is similar 
to a spinning top or gyroscope. This circumstance underlies the 
so-called gyromagnetic phenomena consisting in that the magne- 
tization of a magnetic leads to its rotation, and, conversely, the 
rotation of a magnetic leads to its magnetization. The existence of 
the first phenomenon was proved experimentally by A. Einstein 
and W. de Haas, and of the second by S. Barnett. 

Einstein and de Haas based their experiment on the following 
reasoning. If we magnetize a rod made of a magnetic, then the mag- 
netic moments of the electrons will be aligned in the direction of 
the field, and the angular momenta in the opposite direction. As 
a result, the total angular momentum of the electrons will 

become other than zero (initially owing to the chaotic orientation 
of the individual momenta it equalled zero). The angular momentum 
of the system rod 4- electrons must remain unchanged. Therefore 



168 


Electricity and Magnetism 


the rod acquires the angular momentum — 2 Li and, consequently, 
begins to rotate. A change in the direction of magnetization leads 
to a change in the direction of rotation of the rod. 

A mechanical model of this experiment can be carried out by 
seating a person on a rotatable stool and having him hold a massive 
rotating wheel in his hands. When he holds the axle of 
the wheel upward, he begins to rotate in the direction opposite 
to that of rotation of the wheel. When he turns the 
axle downward, he begins to rotate in the other di- 
rection. 

Einstein and de Haas conducted their experi- 
ment as follows (Fig. 7.11). A thin iron rod was 
suspended on an elastic thread and placed inside a 
solenoid. The thread was twisted very slightly when 
the rod was magnetized using a constant magnetic 
field. The resonance method was used to increase 
the effect — the solenoid was fed with an alternating 
current whose frequency was chosen equal to the 
natural frequency of mechanical oscillations of the 
system. In these conditions, the amplitude of the 
oscillations reached values that could be measured 
by watching the displacement of alight spot reflect- 
ed by a mirror fastened to the thread. The data ob- 
tained in the experiment were used to calculate the gyromagnetic ratio, 
which was found to equal — {elm). Thus, the sign of the charge of 
the carriers setting up the molecular currents coincided with the 
sign of the charge of an electron. The result obtained, however, 
was double the expected value of the gyromagnetic ratio 
(7.41). 

To understand Barnett’s experiment, we must remember that 
when an attempt was made to bring a gyroscope into rotation about 
a certain direction, the gyroscope axis turned so that the directions 
of the natural and forced rotations of the gyroscope coincided (see 
Sec. 5.9 of Vol. I, p. 165 et seq.). If we place a gyroscope fastened 
in a universal joint on the disk of a centrifugal machine and begin 
to rotate it, the gyroscope axis will align itself vertically, and in 
such a way that the direction of rotation of the gyroscope will coin- 
cide with that of the disk. When the direction of rotation of the 
centrifugal machine is reversed, the gyroscope axis will turn through 
180 degrees, i.e. in such a way that the directions of the two rota- 
tions will again coincide. 

Barnett rotated an iron rod very rapidly about its axis and mea- 
sured the produced magnetization. Barnett also obtained a value for 
the gyromagnetic ratio from the results of his experiment 
double that given by Eq. (7.41). 



Magnetic Field in a Substance 


169 


It was discovered later that apart from the orbital magnetic 
moment (7.39) and the orbital angular momentum (7.40), an electron 
has its intrinsic angular momentum L s and magnetic moment p m% s 
for which the gyromagnetic ratio is 


Pm* s e _ 

L b m 


(7.42) 


i.e. coincides with the value obtained in the experiments conducted 
by Einstein and de Haas and by Barnett. It thus follows that the 
magnetic properties of iron are due not to the orbital, but to the 
intrinsic magnetic moment of its electrons. 

Attempts were initially made to explain the existence of the 
intrinsic magnetic moment and angular momentum of an electron 
by considering it as a charged sphere spinning about its axis. Accord- 
ingly, the intrinsic angular momentum of an electron was named 
its spin. It was discovered quite soon, however, that such a notion 
results in a number of contradictions, and it became necessary to 
reject the hypothesis of a “spinning” electron. It is assumed at present 
that the intrinsic angular momentum (spin) and the intrinsic (spin) 
magnetic moment associated with it are inherent properties of an 
electron like its mass and charge. 

Not only electrons, but also other elementary particles have 
a spin. The spin* of elementary particles is an integral or half- 
integral multiple of the quantity h equal to Planck’s constant h 
divided by 2n 

h — -^— = 1.05 x 10" 34 J -s = 1.05 x 10 -27 erg s .(7.43) 


In particular, for an electron, L s = yft; in this connection, the 

spin of an electron is said to equal y. Thus, h is a natural unit of 

the angular momentum like the elementary charge e is a natural 
unit of charge. 

In accordance with Eq. (7.42), the intrinsic magnetic moment 
of an electron is 


Pm — 



e 

m 



eh 

2m 


(7.44) 


The quantity 

Hb = -|£- = 0.927 x 10~ 23 J/T = 0.927 x 10~ 20 erg/Gs** (7.45) 


• More exactly, the maximum value of the pro jection of the spin onto a di- 
rection separated in space, for example, onto that of the external held. 

** According to the equation W= — pmB, the dimension of magnetic moment 
equals that of energy (joule or erg) divided by the dimension of magnetic in- 
duction (tesla or gauss). 


170 


Electricity and Magnetism 


is called the Bohr magneton. Hence, the intrinsic magnetic moment 
of an electron equals one Bohr magneton. 

The magnetic moment of an atom consists of the orbital and intrin- 
sic moments of the electrons in it, and also of the magnetic moment 
of the nucleus (which is due to the magnetic moments of the elemen- 
tary particles — protons and neutrons — 
forming the nucleus). The magnetic mo- 
ment of a nucleus is much smaller than 
the moments of the electrons. For this 
reason, they may be disregarded when 
considering many questions, and we may 
consider the. magnetic moment of an atom 
Fig. 7.12 to e( I ua l the vector sum of the magnetic 

moments of its electrons. The magnetic 
moment of a molecule may also be considered equal to the sum of 
the magnetic moments of all its electrons. 

O. Stern and W. Gerlach determined the magnetic moments of 
atoms experimentally. They passed a beam of atoms through a greatly 
inhomogeneous magnetic field. The inhomogeneity of the field was 
achieved by using a special shape of the electromagnet pole shoes 
(Fig. 7.12). By Eq. (6.77), the atoms 
of the beam must experience the force 


n uu 

r = Pm -fo- cos a 

whose magnitude and sign depend on 
the angle a made by the vector p m 
with the direction of the field. When 
the moments of the atoms are dis- 
tribu ted chaot ica 1 ly by directions, the 
beam contains particles for which 
the values of a vary within the li- 
mits from 0 to jt. It was assumed 
accordingly that a narrow beam of 
atoms after passing between the 
poles would form on a screen a continuous extended trace whose edges 
would correspond to atoms having orientations at angles of a = 0 
and a=ji (Fig. 7.13). The experiment gave unexpected results. 
Instead of a continuous extended trace, separate lines were obtained 
that were arranged symmetrically with respect to the trace of the beam 
obtained in the absence of a field. 

The Stern-Gerlach experiment showed that the angles at which 
the magnetic moments of atoms are oriented relative to a magnetic 
field can have only discrete values, i.e. that the projection of a mag- 
netic moment onto the direction of a field is quantized. 


Without field 


I 


» Expected 

result 

1 Hg.Mg 

1 1 1 Ag.Na.K 

With field I 1 1 I I V 

lllilll Mn 

ii i i i ii ii ft 

Fig. 7.13 




Magnetic Field in a Substance 


174 


The number of possible values of the projection of the magnetic 
moment onto the direction of the magnetic field for different atoms 
is different. It is two for silver, aluminium, copper, and the alkali 
metals, four for vanadium, nitrogen, and the halogens, five for 
oxygen, six for manganese, nine for iron, 
ten for cobalt, etc. 

Measurements gave values of the order of 
several Bohr magnetons for the magnetic 
moments of atoms. Some atoms showed no 
deflections (see, for example, the trace of 
mercury and magnesium atoms in Fig. 7.13), 
which indicates that they have no_ mag- 
netic moment. 


7.7. Diamagiletism 

An electron travelling in an orbit is like a 
spinning top. Therefore, all the features of 
behaviour of gyroscopes under the action of 
external forces must be inherent in it, in 
particular, precession of the electron orbit 
must appear in the appropriate conditions. 

The conditions needed for precession appear 
if an atom is in an external magnetic field B 
(Fig. 7.14). In this case, the torque T = [p m B) 
is exerted on the orbit. It tends to set up 
the orbital magnetic moment of an electron 
p m in the direction of the field (the angular 
momentum L will be set up against the field). 

The torque T causes the vectors p m and L to 
precess about the direction of the magnetic induction vector B whose 
velocity is simple to find (see Sec. 5.9 of Vol. I, p. 165 et seq.). 

During the time dt , the vector L receives the increment 
dL equal to 

dh = T dt 

The vector dh . like the vector T, is perpendicular to the plane 
passing through the vectors B and L; its magnitude is 

| rfL | = p m B sin a dt 

where a is the angle between p m and B. 

During the time dt, the plane containing the vector L will turn 
about the direction of B through the angle 

\dL\ _ P m Hs\nadt Pm D 



172 


Electricity and Magnetism 


Dividing this angle by the time dt , we find the angular velocity of 
precession 

“‘“•ar—i t B 

Introducing the value of the ratio of the magnetic moment and 
angular momentum from Eq. (7.41), we get 

S- <’•«) 

The frequency (7.46) is called the frequency of Larmor precession 
or simply the Larmor frequency. It depends neither on the angle 

of inclination of an orbit with respect to 
the direction of the magnetic field nor on 
the radius of the orbit or the speed of the 
electron, and, consequently, is the same 
for all the electrons in an atom. 

The precession of an orbit causes addi- 
tional motion of the electron about the 
direction of the field. If the distance r' 
from the electron to an axis parallel to B 
and passing through the centre of the orbit 
did not change, the additional motion of 
the electron would occur along a circle 
of radius r' (see the unhatched circle in 
the bottom part of Fig. 7.14). The ring 
current /' = e (co l /2ji) (see the hatched circle) would correspond to 
it. The magnetic moment of this current is 

p' m = I’S’ = e^nr'* = -^r'* (7.47) 

and is directed oppositely to B (see the figure). It is called the in- 
duced magnetic moment. 

Indeed, owing to the motion of an electron in its orbit, the dis- 
tance r' constantly changes. Therefore, in Eq. (7.47), we must replace 
r' 2 with its average value in time <r' 2 ). The latter depends on the 
angle a characterizing the orientation of the orbit plane relative 
to B. In particular, for an orbit perpendicular to the vector B, the 
quantity r' is constant and equals the radius of the orbit r. For 
an orbit whose plane passes through the direction of B, the quantity 
r' varies according to the law r' = r sin o yt, where co is the angular 
velocity of revolution of an electron in its orbit (Fig. 7.15; the vector 
B and the orbit are in the plane of the drawing). Consequently, 

<r' a > = (r 2 sin 2 cat) = -~r 2 (the quantity (sin 2 cat) = . Aver- 

aging over all possible values of a, considering them to be equally 



Magnetic Field in a Substance 


173 


probable, yields 

< r ' 2 > = ■§* r * (7-48) 

Using in Eq. (7.47) the value (7.46) for to L and (7.48) for <r' 2 >, 
we get the following expression for the average value of the induced 
magnetic moment of one electron: 

j ^r*B (7.49) 

(the minus sign reflects the circumstance that the vectors (p^) 
and B have opposite directions). We assumed the orbit to be cir- 
cular. In the general case (for example, for an elliptical orbit), we 
must take (r 2 > instead of r 2 , i.e. the mean square of the distance from 
an electron to the nucleus. 

Summation of Eq. (7.49) over all the electrons yields the induced 
magnetic moment of an atom 

z 

Pm. at = 2 <P*> - - - £r 2 < r *> ( 7 - 5 °) 

(Z is the atomic number of a chemical element; the number of elec- 
trons in an atom is Z). 

Thus, the action of an external magnetic field sets up precession 
of the electron orbits with the same angular velocity (7.46) for all 
the electrons. The additional motion of the electrons due to preces- 
sion leads to the production of an induced magnetic moment of an 
atom {Eq. (7.50)] directed against the field. Larmor precession ap- 
pears in all substances without exception. When atoms by themselves 
have a magnetic moment, however, a magnetic field not only induces 
the moment (7.50), but also has an orienting action on the magnetic 
moments of atoms, aligning them in the direction of the field. The 
positive (i.e. directed along the field) magnetic moment that appears 
may be considerably greater than the negative induced moment. 
The resultant moment is therefore positive, and the substance be- 
haves like a paramagnetic. 

Diamagnetism is found only in substances whose atoms have no 
magnetic moment (the vector sum of the orbital and spin magnetic 
moments of the atom electrons is zero). If we multiply Eq. (7.50) 
by the Avogadro constant N A for such a substance, we get the mag- 
netic moment for a mole of the substance. Dividing it by the field 
strength H y we find the molar magnetic susceptibility Xm.moi- 
The permeability of dielectrics virtually equals unity. We can there- 
fore assume that BtH = p 0 . Thus, 



174 


Electricity and Magnetism 


We must note that the strict quantum-mechanical theory gives 
exactly the same expression. 

Introduction of the numerical values of p 0 , N A , *, and m in 
Eq. (7.51) yields 

z 

Xra, mol = — 3.55 X 10® 2 ( r t) 

The radii of electron orbits have a value of the order of 10~ 10 m. 
Hence, the molar diamagnetic susceptibility of the order of 10“ lt -10~ , 9 
is obtained, which agrees quite well with experimental data. 


7.8. Paramagnetism 

If the magnetic moment p m of the atoms differs from zero, the 
relevant substance is paramagnetic. A magnetic field tends to align 
the magnetic moments of the atoms along B, while thermal motion 
tends to scatter them uniformly in all directions. As a result, a cer- 
tain preferential orientation of the moments is established along 
the field. Its value grows with increasing B and diminishes with 
increasing temperature. 

The French physicist and chemist Pierre Curie (1859-1906) estab- 
lished experimentally a law (named Curie’s law in his honour) 
according to which the susceptibility of a paramagnetic is 

Xm.mol = -f- (7.52) 

where C — Curie constant depending on the kind of substance 
T = absolute temperature. 

The classical theory of paramagnetism was developed by the 
French physicist Paul Langevin (1872-1946) in 1905. We shall limit 
ourselves to a treatment of this theory for not too strong fields and 
not very low temperatures. 

According to Eq. (6.76), an atom in a magnetic field has the 
potential energy W — — p m B cos 0 that depends on the angle 0 
between the vectors p m and B. Therefore, the equilibrium distribu- 
tion of the moments by directions must obey Boltzmann’s law (see 
Sec. 11.8 of Vol. I, p. 326 et seq.). According to this law, the prob- 
ability of the fact that the magnetic moment of an atom will make 
with the direction of the vector B an angle within the limits from 
0 to 0 + d0 is proportional to 

( W \ i p m B cos 0 \ 

-TEf) = ,x P( «■ ) 

Introducing the notation 



(7.53) 



Magnetic Field in a Substance 


175 


W e can write the expression determining the probability in the form 

\ exp (a cos 0) (7.54) 

In the absence of a field, all the directions of the magnetic moments 
are equally probable. Consequently, the probability of the fact that 
the direction of a moment will form with a certain direction z an 
angle within the limits from 0 to 0 + d0 is 

/JD \ d ®o 2ji sin 0 dQ i . A , A 

(dPo)B = 0 = ~4^- = 4^ = “2-sm 0d0 ( / .55) 

Here dQ e = 2n sin 0 d0 is the solid angle enclosed between cones 
having apex angles of 0 and 0 + <^0 
(Fig. 7.16). 

When a field is present, the multiplier 
(7.54) appears in the expression for the 
probability: 

dP$ = A exp (a cos 0) sin 0 d0 (7.56) 

(A is a proportionality constant that is 
meanwhile unknown). 

The magnetic moment of an atom has 
a magnitude of the order of one Bohr 
magneton, i.e. about 10~ 2S J/T [see Eq. (7.45)]. At the usually 
achieved fields, the magnetic induction is of the order of 1 T (10 4 Gs). 
Hence, p m B is of the order of 10~ 23 J. The quantity kT at room tem- 
perature is about 4x 10“ 21 J. Thus, a — p m B/kT is much smaller 
than unity, and exp ( a cos 0) may be replaced with the approximate 
expression 1 + a cos 0. In this approximation, Eq. (7.56) becomes 

dPe = A (1 + a cos 0) sin 0 dQ 

The constant A can be found by proceeding from the fact that 
the sum of the probabilities of all possible values of the angle 0 
must equal unity: 

ji 

1= J A(l + a cos 0) sin 0 dQ == A 
o 

Hence, A = 1, so that 

dPe = -y (1 + a cos 0) sin 0 d0 

Assume that unit volume of a paramagnetic contains n atoms. 
Consequently, the number of atoms whose magnetic moments form 
angles from 0 to 0 4- dQ with the direction of the field will be 

dne = n dP e = ^ n (1 + a cos 0) sin 0 d0 



Fig. 7.16 



170 


Electricity and Magnetism 


Each of these atoms makes a contribution of p m cos 0 to the resultant 
magnetic moment. Therefore, we get the following expression for 
the magnetic moment of unit volume (i.e. for the magnetization): 

n 

M = j p m cos 0 dn 0 — 
o 

K 

= -Y^p m j (1 + a cos 6) cos 0 sin 0 d0 = np m 
o 

Substitution for a of its value from Eq. (7.53) yields 


M 


n P\oP 

3kT 


Finally, dividing M by H and assuming that BlH = p 0 (for a para- 
magnetic (i is virtually equal to unity), we find the susceptibility 


Xm = 


Po"P&> 

3*r 


(7.57) 


Substituting the Avogadro constant for n, we get an expresion 
for the molar susceptibility: 


Xm, mol : 


3kT 


(7.58) 


We have arrived at Curie’s law. A comparison of Eqs. (7.52) and 
(7.58) gives the following expression for the Curie constant: 


3k 


(7.59) 


It must be remembered that Eq. (7.58) has been obtained assuming 
that p m B <£LkT. In very strong fields and at low temperatures, 
deviations are observed from proportionality between the 
magnetization of a paramagnetic M and the field strength H . In 
particular, a state of magnetic saturation may set in when all the 
p m *s are lined up along the field, and a further increase in H does 
not result in a growth in M . 

The values of Xm, moi calculated by Eq. (7.58) in a number of 
cases agree quite well with the values obtained experimentally. 

The quantum theory of paramagnetism takes account of the fact 
that only discrete orientations of the magnetic moment of an atom 
relative to a field are possible. It arrives at an expression for Xm»ruoi 
similar to Eq. (7.58). 


Magnetic Field in a Substance 


177 


7.9. Ferromagnetism 

Substances capable of having magnetization in the absence of 
an external magnetic field form a special class of magnetics. Accord- 
ing to the name of their most widespread representative — ferrum 
(iron) — they hSive been called ferromagnetics. In addition to iron, 
they include nickel, cobalt, gadolinium, their alloys and compounds, 
and also certain alloys and compounds of manganese and chromium 




Fig. 7.18 


with non-ferromagnetic elements. All these substances display fer- 
romagnetism only in the crystalline state. 

Ferromagnetics are strongly magnetic substances. Their magneti- 
zation exceeds that of dia- and paramagnetics which belong to the 
category of weakly magnetized substances an enormous number of 
times (up to 10 10 ). 

The magnetization of weakly magnetized substances varies lin- 
early with the field strength. The magnetization of ferromagnetics 
depends on H in an intricate way. Figure 7.17 shows the magnetiza- 
tion curve for a ferromagnetic whose magnetic moment was ini- 
tially zero (it is called the initial or zero magnetization curve). Already 
in fields of the order of several oersteds (about 100 A/m), the magne- 
tization M reaches saturation. The initial magnetization curve in 
a B-H diagram is shown in Fig. 7.18 (curve 0-1). We remind our 
reader that B = p 0 (H + M). Therefore, when saturation is reached, 
B continues to grow with increasing H according to a linear law: 
B — p 0 £f + const, where const = p 0 ^sat* 

A magnetization curve for iron was first obtained and investigat- 
ed in detail by the Russian scientist Aleksandr Stoletov (1839- 
1896). The ballistic method of measuring the magnetic induction 
which he developed has been finding wide application (see Sec. 8.3). 

Apart from the non-linear relation between H and M (or between 
H and B), ferromagnetics are characterized by the presence of hyster- 
esis. If we bring magnetization up to saturation (point 1 in Fig. 7.18) 


178 


Electricity and Magnetism 


and then diminish the magnetic held strength, the induction B 
will no longer follow the initial curve 0-1 , but will change in accor- 
dance with curve 1-2 . As a result, when the strength of the external 
held vanishes (point 2), the magnetization does not vanish and is 
characterized by the quantity B v called the residual induction. 
The magnetization for this point has the value Af r called the reten- 
tivity or remanence. 

The magnetization vanishes only under the action of the held 
H c directed oppositely to the held that produced the magnetization. 

The held strength H c is called the coer- 
cive force. 

The existence of remanence makes it 
possible to manufacture permanent mag- 
nets, i.e. bodies that have a magnetic mo- 
ment and produce a magnetic held in the 
space surrounding them without the 
expenditure of energy for maintaining the 
macroscopic currents. A permanent mag- 
net retains its properties better when the 
coercive force of the material it is made 
of is higher. 

When an alternating magnetic held acts 
on a ferromagnetic, the induction changes 
in accordance with curve 1-2-3-4-5-1 
(Fig. 7.18) called a hysteresis loop (a simi- 
lar curve is obtained in an M-H dia- 
gram). If the maximum values of H are 
such that the magnetization reaches satu- 
ration, we get the so-called maximum hysteresis loop (the solid loop 
in Fig. 7.18). If saturation is not reached at the amplitude values of 
H y we get a loop called a partial cycle (the dash line in the figure). 
The number of such partial cycles is inhnite, and all of them are 
within the maximum hysteresis loop. 

Hysteresis results in the fact that the magnetization of a ferromag- 
netic is not a unique function of H. It depends very greatly on the 
previous history of a specimen — on the fields which it was in pre- 
viously. For example, in'a field of strength H 1 (Fig. 7.18), the induc- 
tion may have any value ranging from B[ to B \ . 

It follows from everything said above about ferromagnetics that 
they are very similar in their properties to ferroelectrics (see Sec. 2.9). 

In connection with the ambiguity of the dependence of B on H f 
the concept of permeability is applied only to the initial magnetiza- 
tion curve. The permeability of ferromagnetics p (and, consequently, 
their magnetic susceptibility Xm) is a function of the field strength. 
Figure 7.19a shows an initial magnetization curve. Let us draw 
from the origin of coordinates a straight line that passes through an 




Magnetic Field in a Substance 


179 


arbitrary point on the curve. The slope of this line is proportional 
to the ratio B/H, i.e. to the permeability p for the relevant value 
of the field strength. When H grows from zero, the slope (an<J, conse- 
quently, p) first grows. At point 2 it reaches a maximum (straight 
line 0-2 is a tangent to the curve) and then diminishes. Figure 7.196 
shows how p depends on H. A glance at the figure shows that the 
maximum value of the permeability is reached somewhat earlier 
than saturation. Upon an unlimited increase in H , the permeability 
approaches unity asymptotically. This can be seen from the circum- 
stance that M in the expression p — 1 + MIH cannot exceed the 
value A/gat. 

The quantities B r (or M T ), H c . and p ma x are the basic characteristics 
of a ferromagnetic. If the coercive force H c is great, the ferromagnetic 
is called hard. It is characterized by a broad hysteresis loop. A fer- 
romagnetic with a low H c (and accordingly with a narrow hysteresis 
loop) is called soft. The characteristic of a ferromagnetic is chosen 
depending on the use it is to be put to. Thus, hard ferromagnetics 
are used for permanent magnets, and soft ones for the cores of trans- 
formers. Table 7.1 gives the characteristics of several typical fer- 
romagnetics. 


Table 7.1 


Substance 

Composition 



H c , A/m 

Iron 

99.9% Fe 

5 000 


80 

Supermalloy 

79% Ni, 5% Mo, t6% Fe 

800 000 

— 

0.3 

Alniko 

10% Al, 19% Ni, 18% Co. 
53% Fe 


0.9 

52 000 


The fundamentals of the theory of ferromagnetism were presented 
by the Soviet physicist Yakov Frenkel (1894-1952) and the German 
physicist Werner Heisenberg (1901-1976) in 1928. It follows from 
experiments involving the studying of gyromagnetic phenom- 
ena (see Sec. 7.6) that the intrinsic (spin) magnetic moments of 
electrons are responsible for the magnetic properties of ferromagnetics. 
In definite conditions, forces* may appear in crystals that make 
the magnetic moments of the electrons become lined up parallel 
to one another. The result is the setting up of regions of spontaneous 
magnetization, also called domains. Within the confines of each 
domain, a ferromagnetic is spontaneously magnetized to saturation 


* These forces are called exchange ones. Their explanation is given only by 
quantum mechanics. 












180 


Electricity and Magnetism 


and has a definite magnetic moment. The directions of these moments 
are different for different domains (Fig. 7.20), so that in the afbsence 
of an external field the total moment of an entire body is zero. Do- 
mains have dimensions of the order of 1 to 10 p,m. 

The action of a field on domains at different stages of the magneti- 
zation process is different. First, with weak fields, displacement 
of the domain boundaries is observed. As a result, the domains 
whose moments make a smaller angle with H grow 
H V at the expense of the domains for which the angle 

0 between the vectors p m and H is greater. For 
example, domains 1 and 3 in Fig. 7.20 grow at 
the expense of domains 2 and 4 . With an increase 
in the field strength, this process goes on further 
and further until the domains with a smaller 0 
(which have a smaller energy in a magnetic field) 
completely absorb the domains that are less 
advantageous from the energy viewpoint. In the 
next stage, the magnetic moments of the domains 
turn in the direction of the field. The moments 
of the electrons within the confines of a domain 
turn simultaneously without violating their 
strict parallelism to one another. These processes (excluding slight 
displacements of the boundaries between the domains in very weak 
fields) are irreversible, and this is exactly what causes hysteresis. 

There is a definite temperature T c for every ferromagnetic at 
which the regions of spontaneous magnetization (domains) break 
up and the substance loses its ferromagnetic properties. This temper- 
ature is called the Curie point. It is 768 °C for iron and 365 C for 
nickel. At a temperature above the Curie point, a ferromagnetic 
becomes an ordinary paramagnetic whose magnetic susceptibility 
obeys the Curie-Weiss law 

Xm.mol = ( 7 - 60 ) 

(compare with Eq. (7.52)1. When a ferromagnetic is cooled to below 
its Curie point, domains once more appear in it. 

Exchange forces sometimes result in the appearance of so-called 
antiferromagnetics (chromium, manganese, etc.). The existence of 
antiferromagnetics was predicted by the Soviet physicist Lev Lan- 
dau (1908-1968) in 1933. In antiferromagnetics, the intrinsic magnetic 
moments of the electrons are spontaneously oriented antiparallel 
to one another. Such an orientation involves adjacent atoms in 
pairs. The result is that antiferromagnetics have an extremely low 
magnetic susceptibility and behave like very weak paramagnetics. 
There is also a temperature 7 n for antiferromagnetics at which 
the antiparallel orientation of the spins vanishes. This temperature 



Fig. 7.20 




Magnetic Field in a Substance 


181 


is known as the antiferromagnetic Curie point or the Neel point. 
Some antiferromagnetics (for example, erbium, dysprosium, alloys 
of manganese and copper) have two such points (an upper and a lower 
N£el point), the antiferromagnetic properties being observed only 
at the intermediate temperatures. Above the upper point, the sub- 
stance behaves like a paramagnetic, and at temperatures below the 
lower Neel point it becomes a ferromagnetic. 


CHAPTER 8 ELECTROMAGNETIC 

INDUCTION 


8.1. The Phenomenon of Electromagnetic 
Induction 

In 1831, the British physicist and chemist Michael Faraday (1791- 
1867) discovered that an electric current is produced in a closed con- 
ducting loop when the flux of magnetic induction through the sur- 
face enclosed by this loop changes. This phenomenon is called elec- 
tromagnetic induction, and the current produced an induced current. 

The phenomenon of electromagnetic induction shows that when 
the magnetic flux in a loop changes, an induced electromotive force 



Fig. 8.1 

Si is set up. The value of S t does not depend on how the magnetic 
flux <D is changed and is determined only by the rate of change of O, 
i.e. by the value of d<J>/dt . A change in the sign of d<b/dt is attended 
by a change in the direction of 

Let us consider the following example. Figure 8.1 shows loop 1 
whose current I x can be varied by means of a rheostat. This current 
sets up a magnetic field through loop 2. If we increase the current / lf 
the magnetic induction flux <I> through loop 2 will grow. This will 
lead to the appearance in loop 2 of the induced current l % registered 
by a galvanometer. Diminishing of the current I x will cause the mag- 
netic flux through the second loop to decrease. This will result in 
the appearance in it of an induced current of a direction opposite 
to that in the first case. An induced current / 2 can also be set up by 
bringing loop 2 closer to loop 1 or moving it away from it. In these 


Electromagnetic Induction 


183 


two cases, the directions of the induced current are opposite. Finally, 
electromagnetic induction can be produced without translational 
motion of loop 2, but by turning it so as to change the angle between 
a normal to the loop and the direction of the held. 

E. Lenz established a rule permitting us to find the direction of 
an induced current. Lenz’s rule states that an induced current is 
always directed so as to oppose the cause producing it . If, for example, 
a change in 3* is due to motion of loop 2, then an induced current 
is set up of a direction such that the force of interaction with the first 
loop opposes the motion of the loop. When loop 2 approaches loop 
1 (see Fig. 8.1), a current /' is set up whose magnetic moment is 
directed oppositely to the field of the current I x (the angle a between 
the vectors p^andB isji). Hence, loop 2 will experience a force repel- 
ling it from loop 7 [see Eq. (6.77)]. When loop 2 is moved away 
from loop 7, the current I\ is produced whose moment coincides 
in direction with the field of the current I x (a = 0) so that the force 
exerted on loop 2 is directed toward loop 7. 

Assume that both loops are stationary and the current in loop 2 
is induced by changing the current I x in loop 7. Now a current / 2 
is induced of a direction such that the intrinsic magnetic flux it 
produces tends to weaken the change in the external flux leading to 
the setting up of the induced current. When I x grows, i.e. the external 
magnetic flux directed to the right is increased, a current 7' is induced 
that sets up a flux directed to the left. When I x diminishes, the 
current I\ is set up whose intrinsic magnetic flux has the same direc- 
tion as the external flux and, consequently, tends to keep the external 
flux unchanged. 


8.2. Induced E.M.F. 

We have established in the preceding section that changes in the 
magnetic flux <J> through a loop set up an induced e.m.f. in it. 
To find the relation between and the rate of change of O, we shall 
consider the following example. 

Let us take a loop with a movable rod of length l (Fig. 8.2a). 
We shall put it in a homogeneous magnetic field at right angles 
to the plane of the loop and directed beyond the drawing. Let us 
bring the rod into motion with the velocity v. The current carriers 
in the rod — electrons — will also begin to move relative to the field 
with the same velocity. As a result, each electron will begin to expe- 
rience the magnetic force 

F, = — <?[vBl (8.1) 

directed along the rod [see Eq. (6.33); the charge of an electron is — eh 
The action of this force is equivalent to the action on an electron 



184 


Electricity and Magnetism 


of an electric field of strength 

E = [vBl 

This field is of a non-electrostatic origin. Its circulation around 
a loop gives the value of the e.m.f. induced in the loop: 

2 

[vB]dl= j [vB]dl (8.2) 

i 

(the integrand differs from zero only on section 1-2 formed by the rod). 



Fig. 8.2 


To be able to judge about the direction in which the e.m.f. acts 
according to the sign of <& u we shall consider positive when its 
direction forms a right-handed system with the direction of a normal 
to the loop. 

Let us choose the normal as shown in Fig. 8.2. Hence, when cal- 
culating the circulation, we must circumvent the loop clockwise 
and choose the direction of the vectors d\ accordingly. If we ‘put 
the constant vector [vB] in Eq. (8.2) outside the integral, we get 

2 

g, = IvB] j dl = [vB] 1 

1 

where 1 is the vector depicted in Fig. 8.26. Let us perform a cyclic 
rearrangement of the multipliers in the expression obtained, after 
which we shall multiply and divide it by dt: 

= B [lv] = Bll, d - V --L (8.3) 

A glance at Fig. 8.26 shows that 

[1, v dt] — — n dS 

where dS is the increment of the loop area during the time dt. By 
the definition of a flux, B dS ^ Bn dS is the flux through the 


Electromagnets Induction 


185 


area dS , i.e. the increment of the flux dO through the loop. Thus, 
B II, v dt] = —Bn dS = —d( D 
With a view to this expression, Eq. (8.3) can be written as 

d<P 




dt 


(8.4) 


B® 


Text 


We have found that dO/dJ and have opposite signs. The sign 
of the flux and that of are associated with the choice of the direc- 
tion of a normal to the plane of a loop. With 
our selection of the normal (see Fig. 8.2), the 
sign of d<t>/dt is positive, and that of gj is neg- 
ative. If we had chosen a normal directed 
not beyond the drawing, but toward us, the 
sign of dO/df would be negative and that of % t 
positive. 

The SI unit of magnetic induction flux is the 
weber (Wb), which is the flux through a sur- 
face of 1 m 2 intersected by magnetic field lines 
normal to it with B = 1 T. At a rate of change of 
the flux equal to 1 Wb/s, an e.m.f. of f V is induced in the 
In the Gaussian system of units, Eq. (8.4) has the form 

<*, 1 dO 

g ‘ =— T"5T 



Fig. 8.3 


loop. 


(8.5) 


The unit of <t> in this system is the maxwell (Mx) equal to the flux 
through a surface of 1 cm 2 at B = 1 Gs. Equation (8.5) gives 
in cgsey. To find it in volts, we must multiply the result obtained 
by 300. Since 300 lc = 10 -8 , we have 

Si (V) = - 10- 8 (Mx/s) (8.6) 

In the reasoning that led us to Eq. (8.4), the part of the extraneous 
forces maintaining a current in a loop was played by magnetic 
forces. The work of these forces on a unit positive charge, equal by 
definition to the e.m.f., is other than zero. This circumstance appar- 
ently contradicts the statement made in Sec. 6.5 that a magnetic 
force can do no work on a charge. This contradiction is eliminated 
if we take into account that the force (8.1) is not the total magnetic 
force exerted on an electron, but only the component of this force 
parallel to the conductor and due to the velocity v (see the force 
F B in Fig. 8.3). This component causes the electron to start moving 
along the conductor with the velocity u, as a result of which a mag- 
netic force perpendicular to the wire is set up equal to 

F x = — e[uB] 

(this component makes no contribution to the circulation because 
it is perpendicular to dl). 


186 


Electricity and Magnetism 


The total magnetic force exerted on an electron is 

F=F| + F X 

and the work of this force on an electron during the time dt is 

dA = F | u dt + F ± v dt = F 0 u dt—'F x u dt 

(the directions of the vectors F fl and u are the same, and of the 
vectors Fj_ and v are opposite; see Fig. 8.3). Substituting for the 
magnitudes of the forces their values F jj = evB and F x = euB, 
we find that the work of the total magnetic force equals zero. 

The force Fj^ is directed oppositely to the velocity of the rod v. 
Therefore, for the rod to move with the constant velocity v, the external 
force F ex t must be applied to it that balances the sum of the forces 
F ± applied to all the electrons contained in the rod. It is exactly 
at the expense of the work of this force that the energy liberated 
in the loop by the induced current will be produced. 

Our explanation of the appearance of an induced e.m.f. relates 
to the case when the magnetic field is constant, while the geometry 
of the loop changes. The magnetic flux through the loop can also 
be changed, however, by changing B. In this case, the explanation 
of the appearance of an e.m.f. will differ in principle. The time-varying 
magnetic field sets up a vortex electric field E (this is treated in 
detail in Sec. 9.1). The action of the field E causes the current carriers 
in a conductor to start moving — an induced current is set up. The 
relation between the induced e.m.f. and the changes in the magnetic 
flux in this case too is described by Eq. (8.4). 

Assume that the loop in which an e.m.f. is induced consis ts of N 
turns instead of one, i.e. it is a solenoid, for example. Since the turns 
are connected in series, will equal the sum of the e.m.f. ’s induced 
in each of the turns separately: 

Sf-4(2») 

The quantity 

<*> (8.7) 

is called the flux linkage or the total magnetic flux. It is measured 
in the same units as O. If the flux through each of the turns is the 
same, then 

Y-MD (8.8) 

The e.m.f. induced in an intricate loop is determined by the for- 
mula 


_ 

1 “ dt 


(8.9) 



Electromagnetic Induction 


187 


8.3. Ways of Measuring 
the Magnetic Induction 

Assume that the total magnetic flux linked to a loop changes 
from ¥1 to V t . Let us find the charge q that flows through each sec- 
tion of the loop. The instantaneous value of the current in the loop is 

r 81 _ 1 d? 

R R it 

Hence, 

Integration of this expression yields the total charge: 

2 

? = j dq= j (8 .i0) 

1 

Equation (8.10) underlies the ballistic method of measuring the 
magnetic induction developed by A. Stoletov. It consists in the 



following. A small coil with N turns is placed in the field being 
studied. The coil is arranged so that the vector B is perpendicular 
to the plane of the turns (Fig. 8.4a). Hence, the total magnetic flux 
linked with the coil will be 

= NBS 

where S is the area of one turn, which must be so small that the field 
within its limits may be considered homogeneous. 

When the coil is turned through 180 degrees (Fig. 8.46), the flux 
linkage becomes equal to = — NBS (n and B are directed oppo- 
sitely). Hence, the change in the total flux linkage when the coil 
is turned is Y, — = 2NBS . If t.b« mil 



188 


Electricity and Magnetism 


quickly, a short current- pulse is produced in the loop upon which 
the charge 

q = -j r 2NBS (8.11) 

flows [see Eq. (8.10)1. 

The charge flowing in the circuit during the short current pulse 
can be measured with the aid of a so-called ballistic galvanometer. 
The latter is a galvanometer with a great period of natural oscilla- 
tions. Having measured q and knowing R , N , and S , we can find B 

by Eq. (8.11). By R here is meant the re- 
sistance of the entire circuit including the 
coil, the connecting wires, and the gal- 
vanometer. 

Instead of turning the coil, we may 
switch on (or off) the magnetic field being 
studied, or reverse its direction. 

To measure B, the circumstance is also 
used that the electric resistance of bismuth 
grows greatly under the action of a magnetic field — by about five per 
cent per tenth of a tesla (per 1000 Gs). Consequently, we can deter- 
mine the magnetic induction of a magnetic field by placing a prelimi- 
narily graduated bismuth coil (Fig. 8.5) into the field and measuring 
the relative change in its resistance. 

We must note that the electric resistance of other metals also 
grows in a magnetic field, but to a much smaller extent. For copper, 
for example, the increase in the resistance is about one-ten thousandth 
of that for bismuth. 



8.4. Eddy Currents 

Induced currents can also be produced in solid massive conductors. 
In this case, they are known as eddy currents. The electric resistance 
of a massive conductor is small, therefore the eddy currents may 
reach a very high value. 

In accordance with Lenz’s rule, eddy currents choose paths and 
directions in a conductor such as to resist by their action the reason 
setting them up as much as possible. This is why good conductors 
moving in a strong magnetic field experience great retardation due 
to the interaction of the eddy currents with the magnetic field. This 
is taken advantage of for damping the movable parts of galvano- 
meters, seismographs, and other instruments. A conducting (for 
example, aluminium) plate in the form of a sector is fastened to 
the movable part of an instrument (Fig. 8.6) and is introduced into 
the sraD between the poles of a strong permanent magnet. Movement 



Electromagnetic Induction 


189 


of the plate causes eddy currents to be produced in it that brake 
the system. The advantage of such a device is that the braking action 
appears only when the plate moves and vanishes when the plate 
is stationary. Therefore, the electromagnetic damper is absolutely 
no hindrance to the instrument accurately arriving at its equilibrium 
position. 

The thermal action of eddy currents is used in induction furnaces. 
Such a furnace is a coil supplied with a high-frequency current of 
a high value. If we place a conducting body inside the coil, intensive 
eddy currents will be produced in it that 
can heat the body up to its melting point. 

This method is used to melt metals in a va- 
cuum. The resulting materials have an ex- 
ceedingly high purity. 

Eddy currents are also used to heat the 
internal metal components of vacuum instal- 
lations in order to degas them. 

Eddy currents are quite often undesir- 
able, and special measures must be taken to 
eliminate them. For example, to prevent the 
losses of energy for heating transformer cores 
by eddy currents, the cores are assembled of 
thin insulated sheets. The latter are arranged 

so that the possible directions of the eddy currents will be perpendicu- 
lar to them. The appearance of ferrites (semiconductor magnetic ma- 
terials with a high electric resistance) made it possible to manufacture 
solid cores. 

The eddy currents set up in conductors carrying alternating currents 
are directed so as to weaken the current inside a conductor and increase 
it near the surface. As a result, the fast-varying current is distrib- 
uted unevenly over the cross section of the conductor — it is forced 
out, as it were, to the surface of the conductor. This phenomenon 
is called the skin effect. Owing to this effect, the internal part of 
conductors in high-frequency circuits is useless. This is why the 
conductors used for such circuits have the form of tubes. 



8.5. Self-Induction 

An electric current flowing in any loop produces the magnetic flux 
Y through this loop. When / changes, W also changes, and the result 
is the induction of an e.m.f. in the loop. This phenomenon is called 
self-induction. 

In accordance with the Biot-Savart law, the magnetic induction B 
is proportional to the current setting up the field. Hence, it follows 
that the current / in a loop and the total magnetic flux V thrmiorh 



190 


Electricity and Magnetism 


the loop it produces are proportional to each other: 

V = LI (8.12) 

The constant of proportionality L between the current and the 
total magnetic flux is called the inductance ofa loop. 

A linear dependence of on I is observed only if the permeability 
p of the medium surrounding the loop does not depend on the field 
strength H , i.e. in the absence of ferromagnetics. Otherwise, p 
is an intricate function of / (through H , see Fig. 7.19&), and, since 
B =p 0 p H, the dependence of V on I will also be quite intricate. 
Equation (8.12), however, is also extended to this case, and the induc- 
tance L is considered as a function of I. With a constant current /, 
the total flux Y can change as a result of changes in the shape and 
dimensions of a loop. 

It can be seen from the above that the inductance L depends 
on the geometry of a loop (i.e. on its shape and dimensions), and 
also on the magnetic properties (on p) of the medium surrounding 
the loop. If the loop is rigid and there are no ferromagnetics near 
it. the inductance L is a constant quantity. 

The SI unit of inductance is the inductance of a conductor in 
which a total flux ¥ of 1 Wb linked with it is set up at a current 
of 1 A in the conductor. This unit is called the henry (H). 

In the Gaussian system of units, the inductance has the dimension 
of length. Accordingly, the unit of inductance in this system is 
called the centimetre. A loop with which a flux of 1 Mx (10“ 8 Wb) 
is linked at a current of 1 cgsmj (i.e. 10 A) has an inductance of 1 cm. 

Let us calculate the inductance of a solenoid. We shall take a sole- 
noid so long that it can virtually be considered infinite. When 
a current / flows in it, a homogeneous field is produced inside the 
solenoid whose induction is B = p 0 p nl [see Eqs. (6.108) and (7.26)1. 
The flux through each of the turns is O = BS, and the total magnetic 
flux linked with the solenoid is ^ 

¥ = N<X>=nlBS = ii 0 im*lSI (8.13) 

where l = length of the solenoid (which is assumed to be very great) 
S = cross-sectional area 

t n = number of turns per unit length (the product nl gives 
the total number of turns N). 

A comparison of Eqs. (8.12) and (8.13) gives the following expres- 
sion for the inductance of a very long solenoid: 

L = p 0 pn 2 Z5 = p 0 pn 2 F (8. 14) 

where V = IS is the volume of the solenoid. 

It follows from Eq. (8.14) that the dimension of p 0 equals that 
of inductance divided by the dimension of length. Accordingly. 
>• fa moacnrorl ir» henrv Der metre [see Eq. (6.3)1. 


Electromagnetic Induction 


191 


When the current in a loop changes, a self-induced e.m.f. is 
set up that equals 


dY _ d (LI) _ / , dl T dL \ 

dt dt ~~ dt ■ dt ) 


(8.15) 


If the inductance remains constant when the current changes (which 
is possible only in the absence of ferromagnetics), the expression 
for the self-induced e.m.f. becomes 


< 8 - 16) 

The minus sign in Eq. (8.16) is due to Lenz’s rule according to which 
an induced cunent is directed so as to oppose the cause producing 
it. In the case being considered, what sets up g 8 is the change of 
the current in the circuit. Let us assume clockwise circumvention 
to be the positive direction. In these conditions, the current will 
be greater than zero if it flows clockwise in the circuit and less than 
zero if it flows counterclockwise. Similarly, % B will be greater, than 
zero if it is exerted in a clockwise direction, and less than zero if it 
is exerted in a counterclockwise one. 

The derivative dl/dt is positive in two cases — either upon a growth 
in a positive current or upon a decrease in the absolute value of 
a negative current. Inspection of Eq. (8.16) shows that in these cases 
< 0. This signifies that the self-induced e.m.f. is directed counter- 
clockwise and, therefore, is opposed to the above current changes (a 
growth in a positive or a decrease in a negative current). 

The derivative dlldt is negative also in two cases — either when 
a positive current diminishes, or when the magnitude of a negative 
current grows. In these cases, g s >0 and, consequently, opposes 
changes in the current (a decrease in a positive or a growth in the 
magnitude of a negative current). 

Equation (8.16) makes it possible to define the inductance as 
a constant of proportionality between the rate of change of the 
current in a loop and the resulting self-induced e.m.f. Such a defini- 
tion is lawful, however, only when L — const. In the presence of 
ferromagnetics, L of an undeforming loop will be a function of I 
(through H ). Hence, dL/dt can be written as (< dL/dI)(dl!dt ). Making 
such a substitution in Eq. (8.15), we get 

*■ ( i + / -^-)-§- < 8 - 17 > 

We can thus see that in the presence of ferromagnetics the constant 
of proportionality between dlldt and does not at all equal Z. 



192 


Electricity and Magnetism 


8.6. Current When a Circuit 
Is Opened or Closed 


According to Lenz’s rule, the additional currents set up owing 
to self-induction are always directed so as to prevent any changes 
in the current in a circuit. The result is that a current grows to its 
steady value when a circuit is closed or drops to zero when the circuit 
is opened not instantaneously, but gradually. 

Let us first find how a current changes when the 
switch of a circuit is opened. Assume that a current 
source of e.m.f. % is connected in a circuit with an 
inductance L not depending on /and a resistance R 
(Fig. 8.7). The steady current flowing in the circuit 
will be 

(8.18) 



I -- i. 

ifl 77" 


(we consider the resistance of the current source to 
be negligibly small). 

Fig. 8.7 At the moment t = 0, let us switch off the current 
source and simultaneously short the circuit by means 
of switch Sw . As soon as the current in the circuit begins to dim- 
inish, a self-inductance e.m.f. opposing this decrease appears. The 
current in the circuit will comply with the equation 




or 


4 + 4 '=° 


(8.19) 


Equation (8.19) is a linear homogeneous differential equation of the 
first order. Separating variables, we get 


whence 



In / 27” *+ * n const 


(with a view to further transformations, we have written the integra- 
tion constant in the form “In const”). Converting this relation to 
a power yields 

I = const x exp ^ — j- *) (8.20) 

Equation (8.20) is a general solution of Eq. (8.19). We shall find 
the value of the constant from the initial conditions. When t = 0, 
the current had the value given by Eq. (8.18). Hence, const = / 0 - 



Electromagnetic Induction 


193 


Introducing this value into Eq. (8.20), we arrive at the expression 

/ = / 0 exp( — -£-<) (8.21) 

Thus, after the e.m.f. source had been switched off, the current 
in the circuit did not vanish instantaneously, but diminished accord- 
ing to the exponential law (6.21). A plot of the diminishing of / 
is given in Fig. 8.8 (curved). The rate of di- 
minishing is determined by the quantity 

t = -§- (8.22) 

having the dimension of time and called the 
time constant of the circuit. Substituting 
1 /t for RIL in Eq. (8.21), we get 

/ = / 0 exp (--£-) (8.23) 

According to this equation, t is the time during which the current 
diminishes to 1/e-th of its initial value. A glance at Eq. (8.22) shows 
that the time constant t grows and the current in the circuit dimin- 
ishes at a slower rate with an increasing inductance L and a decreas- 
ing resistance R of the circuit. 

To simplify our calculations, we considered that the circuit is short- 
ed when the current source is switched off. If we simply break a cir- 
cuit with a high inductance, the high induced voltage set up produces 
a spark or an arc at the place of breaking of the circuit. 

Now let us consider the closing of a circuit. After the e.m.f. source 
is switched on, a self-induced e.m.f. will act in the circuit apart 
from the e.m.f. % until the current reaches its steady value given 
by Eq. (8.18). Hence, in accordance with Ohm’s law 

or 

J 5T-+-T- / =T‘ < 8 - 24 > 

We have arrived at a linear inhomogeneous differential equation 
that differs from Eq. (8.19) only in that the right-hand side contains 
the constant quantity % IL instead of zero. It is known from the 
theory of differential equations that the general solution of a linear 
inhomogeneous equation can be obtained by adding any partial solu- 
tion of it to the general solution of the corresponding homogeneous 
equation (see Sec. 7.4 of Vol. I, p. 192). The general solution of 
our homogeneous equation has the form of Eq. (8.20). It is easy to 
see that / = %!R = / 0 is a partial solution of Eq. (8.24). Hence. 



Fig. 8.8 



194 


Electricity and Magnetism 


the function 

I = /,»+ const x exp ^ — j- t ) 

will be the general solution of Eq. (8.24). At the initial moment, 
the current is zero. Thus, const = — I 0 , and 

i — [l — exp ( — j- t )J (8.25) 

This function describes the growth of the current in a circuit after 
a source of an e.m.f. has been switched on in it. A plot of function 
(8.25) is shown in Fig. 8.8 (curve 2). 


8.7. Mutual Induction 

Let us take two loops 1 and 2 close to each other (Fig. 8.9). If 
the current I x flows in loop 1 , it sets up through loop 2 a total magnetic 
flux proportional to I x , i.e. 

Y 2 = L 21 / t (8.26) 

(the field producing this flux is depicted in the figure by solid lines). 
When the current I x changes, the e.m.f. 

S..2=-£ 2 i-^ (8.27) 



is induced in loop 2 (we assume that there are no ferromagnetics 
near the loops). 

Similarly, when the current / 2 flows in loop 2 , the following flux 
linked with loop 1 appears: 

^ = ^7, (8.28) 


(the field producing this flux is depicted in the figure 
When the current 7* changes, the e.m.f. 


*i., 


by dash lines). 


is induced in loop 1 . 


(8.29) 


Electromagnetic Induction 


195 


Loops 1 and 2 are called coupled, while the phenomenon of the 
setting up of an e.m.f. in one of the loops upon changes in the current 
in the other is called mutual induction* 

The coefficients of proportionality Lit and L 21 are called the 
mutual inductances of the loops. The relevant calculations show 
that in the absence of ferromagnetics these 
coefficients are always equal to each other: 

L i2 = L 2i (8.30) 

Their magnitude depends on the shape, di- 
mensions, and mutual arrangement of the 
loops, and also on the permeability of the 
medium surrounding the loops. The quantity 
L 12 is measured in the same units as the 
inductance L. 

Let us find the mutual inductance of two 
coils wound onto a common toroidal iron core 
(Fig. 8.10). The magnetic induction lines 
are concentrated inside the core [see the 
text following Eq. (7.37)]. We can therefore 
consider that the magnetic field set up by 
any of the windings will have the same strength throughout the core. 
If the first winding has N 1 turns and the current I x flows through 
it, then according to the theorem on circulation [see Eq. (7.11)1, 
we have 

HI = N X I X (8.31) 

(here l is the length of the core). 

The magnetic flux through the cross section of the core is O = 
= BS = p 0 p/LS, where S is the cross-sectional area of the core. 
Introducing the value of H from Eq. (8.31) and multiplying the 
expression obtained by N 2y we get the total flux linked with the 
second winding: 

A comparison of this equation with Eq. (8.26) shows that 

( 832 > 

Calculations of the flux Y, linked with the first winding when the 
current /, flows through the second winding yields the equation 

L i2 = j- IhpNiMi (8.33) 

which coincides in form with L 21 [see Eq. (8.32)]. In the given case, 
however, we cannot assert that L 12 = L 21 . The factor p in the ex- 



Fig. 8.10 


196 


Electricity and Magnetism 


pressions for these coefficients depends on the field strength H in 
the core. If N x then the same current passed once through the 

first winding and another time through the second one will set up 
a field of different strength H in the core. Accordingly, the values of 
p in both cases will be different so that when I x — I t the numerical 
values of Lit and L 2 1 do not coincide. 


8.8. Energy of a Magnetic Field 

Let us consider the circuit shown in Fig. 8.11. When the switch 
is closed, the current /will be set up in the solenoid. It will produces 
magnetic field linked with the solenoid turns. If the switch is opened, 

a gradually diminishing current 
will flow for a certain time through 
resistor R. This current is maintained 
by the self-induced e.m.f. produced in 
the solenoid. The work done by the cur- 
rent during the time dt is 

dA= % t I dt = dW 





<n 


■I dt= — I d*P 


r tV- 


n A-. 


V 


Fig. 8.11 


(8.34) 

If the inductance of the solenoid does 
not depend on I (L — const), then 
cP¥ = L dl, and Eq. (8.34) becomes 

dA = —LI dl (8.35) 


Integrating this expression with respect to I within the limits from 
the initial value of I to zero, we get the work done in the circuit 
during the entire time needed for vanishing of the magnetic field: 


A= — j LI dl = 


(8.36) 


The work (8.36) is spent on an increment of the internal energy 
of the resistor R, the solenoid, and the connecting wires (i.e. on 
heating them). This work is attended by vanishing of the magnetic 
field that initially existed in the space surrounding the solenoid. 
Since no other changes occur in the bodies surrounding the circuit, 
it remains for us to conclude that the magnetic field is a carrier 
of energy, and it is exactly at the expense of the latter that the work 
given by Eq. (8.36) is done. We thus arrive at the conclusion that 
a conductor of inductance L carrying the current / has the energy 

W = (8.37) 



Electromagnetic Induction 


197 


that is localized in the magnetic held set up by the current [compare 
this equation with the expression ClP/2 for the energy of a charged 
capacitor; see Eq. (4.5)1. 

Equation (8.36) can be interpreted as the work that must be done 
against the self-induced e.m.f. when the current grows from 0 to I, 
and that is used to set up a magnetic held having the energy given 
by Eq. (8.37). Indeed, the work done against the self-induced e.m.f. is 

i 

o 

Performing transformations similar to those which led us to Eq. (8.35), 
we get 

i 

■4' = | LI dl = —£~ (8.3 8) 

0 

that coincides with Eq. (8.36). The work according to Eq. (8.38) 
is done when the current sets in at the expense of the e.m.f. source. 
It is used completely for producing a magnetic field linked with 
the solenoid turns. Equation (8.38) takes no account of the work 
spent by the e.m.f. source for heating the conductors during the 
time the current reaches its steady value. 

Let us express the energy of a magnetic field given by Eq. (8.37) 
through quantities characterizing the field itself. For a long (virtually 
infinite) solenoid 

L=Popn 2 V, H = nl\ or /=~ 


[see Eqs. (8.14) and (7.29)1. Using these values of L and I in Eq. (8.37) 
and performing the relevant transformations, we obtain 


w=m£L v 


(8.39) 


It was shown in Sec. 6.12 that the magnetic field of an infinitely 
long solenoid is homogeneous and differs from zero only inside the 
solenoid. Hence, the energy according to Eq. (8.39) is localized inside 
the solenoid and is distributed over its volume with a constant den- 
sity w that can be found by dividing W by V. This division yields 


w 


Mg! 

2 


(8.40) 


Using Eq. (7.17), we can write the equation for the energy density 
of a magnetic field as follows: 

Poll#* HB B* 


w 


9 


9 


2il~.il 


( 8 . 41 ) 



198 


Electricity and Magnetism 


The expressions we have obtained for the energy density of a mag- 
netic field differ from Eqs. (4.11) for the energy density of an electric 
field only in that the electrical quantities in them have been replaced 
with the relevant magnetic ones. 

Knowing the density of the field energy at every point, we can 
find the energy of the field enclosed in any volume V . For this pur- 
pose, we must calculate the integral 


W= j u>dV= j 

V V 


™f-dv 


(8.42) 


It can be shown that for coupled loops (in the absence of ferromag- 
netics) the field energy is determined by the equation 


W 


hli 

2 


2 2 *2 


(8.43) 


A similar expression is obtained for the energy of N loops coupled 
to one another: 

N 

W = ±- S L t . h I t I h (8.44) 

*=1 

where L i%h = L kt * is the mutual inductance of the i-th and A:-th loops, 
and L it t = L t is the inductance of the i-th loop. 


8.9. Work in Magnetic Reversal 
of a Ferromagnetic 

Changes in a current in a circuit are attended by the performance 
of work against the self-induced e.m.f.: 

dA’ = ( — g 8 ) I dt = / dt= I dV (8.45) 

If the inductance of the circuit L remains constant (which is possible 
only in the absence of ferromagnetics), this work is used completely 
for producing the energy of a magnetic field: dA ' = dW. We shall 
now see that matters are different when ferromagnetics are present. 
For a very long (“infinite”) solenoid, H = nl y V = nlBS. Hence, 

d¥ = nlS dB 

Introducing these expressions into Eq. (8.45), we get 

dA' = H dB- V (8.46) 

where V — IS is the volume of the solenoid, i.e. the volume in which 

fl hnmn^PTlAOim m « orn Dtir* Vioo koon nrArln^a/1 


Electromagnetic Induction 


199 


Let us see whether we can identify Eq. (8.46) with the increment 
of the energy of a magnetic field. We remind our reader that energy 
is a function of state. Therefore, the sum of its increments for a cyclic 
process is zero: 

- 0 


If we fill a solenoid with a ferromagnetic, then the relation between 
B and H is depicted by the curve shown in Fig. 8.12. The expres- 
sion H dB gives the area of the hatched 


strip. Consequently, the integral ^ H dB 

calculated along the hysteresis loop equals 
the area S t enclosed by the loop. Thus, 
the integral of expression (8.46), i.e. 

differs from zero. It therefore fol- 
lows that in the presence of ferromagne- 
tics, the work given by Eq. (8.46) cannot 
be equated to the increment of the energy 
of a magnetic field. Upon completion of 
the cycle of magnetic reversal, H and B 
and, therefore, the magnetic energy will 
have their initial values. Hence, the work 



<jpdA' is not used to produce the energy of a magnetic field. Exper- 
iments show that it is used to increase the internal energy of the 
ferromagnetic, i.e. to heat it. 

Thus, the completion of one cycle of magnetic reversal of a ferro- 
magnetic is attended by the expenditure of work per unit volume 
numerically equal to the area of the hysteresis loop: 


*^u.yoi * ^ H dB = S\ 


(8.47) 


This work goes to heat the ferromagnetic. 

In the absence of ferromagnetics, B is an unambiguous function 
of H (B = p 0 p/7, where p = const). Therefore, the expression 
H dB — pop/T dH is a total differential 

dw = H dB (8.48) 


determining the increment of the energy of a magnetic field. Integra- 
tion of Eq. (8.48) within the limits from 0 to H leads to Eq. (8.40) 
for the density of the field energy (before performing integration, 
H dB must be transformed by substituting p 0 pd/7 for dB). 



CHAPTER 9 MAXWELL’S EQUATIONS 


9.1. Vortex Electric Field 

Let us consider electromagnetic induction when a wire loop in 
which a current is induced is stationary, and the changes in the mag- 
netic flux are due to changes in the magnetic field. The setting up 
of an induced current signifies that the changes in the magnetic 
field produce extraneous forces in the loop that are exerted on the 
current carriers. These extraneous forces are associated neither with 
chemical nor with thermal processes in the wire. They also cannot be 
magnetic forces because such forces do not work on charges. It 
remains to conclude that the induced current is due to the electric 
field set up in the wire. Let us use the symbol E B to denote the 
strength of this field (this symbol, like the one E g used below, is 
an auxiliary one; we shall omit the subscripts B and q later on). 
The e.m.f. equals the circulation of the vector E B around the given 
loop: 

£i = <^E B dl (9.1) 

Introducing into the equation = — d<X>/dt Eq. (9.1) for 

and the expression ( BdS for d> t we arrive at the equation 

s 

(the integral in the right-hand side of the equation is taken over 
an arbitrary surface resting on the loop). Since the loop and the 
surface are stationary, the operations of time differentiation and 
integration over the surface can have their places exchanged: 

<§E B dl =- f -|2-dS (9.2) 

In connection with the fact that the vector B depends, generally 
speaking, both on the time and on the coordinates, we have put 
the symbol of the partial time derivative inside the integral (the 

integral f B dS is a function only of time). 




Maxwell' t Equations 


201 


Let us transform the left-hand side of Eq. (9.2) in accordance with 
Stokes’s theorem. The result is 

j[VE J ,]dS=-|-g-dS 

Owing to the arbitrary nature of choosing the integration surface, 
the following equation must be obeyed: 

lVE fl ] = ~~ (9.3) 

The curl of the field E B at each point of space equals the time deriv- 
ative of the vector B taken with the opposite sign. 

The British physicist James Maxwell (1831-1879) assumed that 
a time-varying magnetic field causes the field E B to appear in space 
regardless of whether or not there is a wire loop in this space. The 
presence of a loop only makes it possible to detect the existence of 
an electric field at the corresponding points of space as a result of 
a current being induced in the loop. 

Thus, according to Maxwell’s idea, a time-varying magnetic 
field gives birth to an electric field. This field E B differs appreciably 
from the electrostatic field E g set up by fixed charges. An electrostatic 
field is a potential one, its strength lines begin and terminate at 
charges. The curl of the vector Eg is zero at any point: 

[VE g ] = 0 (9.4) 

[see Eq. (1.112)]. According to Eq. (9.3), the curl of the vector E B 
differs from zero. Hence, the field E B , like a magnetic field, is a vor- 
tex one. The strength lines of the field E B are closed. 

Thus, an electric field may be either a potential (Eg) or a vortex 
(E B ) one. In the general case, an electric field can consist of the 
field Eg produced by charges and the field E B set up by a time- 
varying magnetic field. Adding Eqs. (9.4) and (9.3) v we get the fol- 
lowing equation for the curl of the strength of the total field E = 
= Eg + E b : 

[VE]=-4*L (9.5) 

This equation is one of the fundamental ones in Maxwell’s electro- 
magnetic theory. 

The existence of a relationship between electric and magnetic 
fields [expressed in particular by Eq. (9.5)1 is a reason why the sepa- 
rate treatment of these fields has only a relative meaning. Indeed, 
an electric field is set up by a system of fixed charges. If the charges 
are fixed relative to a certain inertial reference frame, however, they 
are moving relative to other inertial frames and, consequently, 
set up not only an electric, but also a magnetic field. A stationary 
wire carrying a steady current sets up a constant magnetic field 


202 


Electricity and Magnetism 


at every point of space. This wire is in motion, however, relative 
to other inertial frames. Consequently, the magnetic field it sets 
up at any point with the given coordinates x, y, z will change and 
therefore give birth to a vortex electric field. Thus, a field which 
is “purely” electric or “purely” magnetic relative to a certain reference 
frame will be a combination of an electric and a magnetic field 
forming a single electromagnetic field relative to other reference 
frames. 


9.2. Displacement Current 


For a stationary (i.e. not varying with time) electromagnetic 
field, the curl of the vector H by Eq. (7.9) equals the density of the 
conduction current at each point: 

[VH] - j 

The vector j is associated with the charge density at the same point 
by continuity equation (5.11): 


Vj — — 


ap 

dt 


An electromagnetic field can be stationary only provided that 
the charge density p and the current density j do not depend on the 



time. In this case, according to Eq. (5.11), the divergence of j equals 
zero. Therefore, the current lines (lines of the vector j) have no 
sources and are closed. 

Let us see whether Eq. (7.9) holds for time-varying fields. We shall 
consider the current flowing when a capacitor is charged from a source 
of constant voltage U. This current varies with time (the current 
stops flowing when the voltage across the capacitor becomes equal 
to U ). The lines of the conduction current are interrupted in the 
space between the capacitor plates (Fig. 9.1; the current lines inside 
the plates are shown by dash lines). 



Max weir 9 Equations 


203 


Let us take a circular loop T enclosing the wire in which the current 
flows toward the capacitor and integrate Eq. (7.9) over surface S x 
intersecting the wire and enclosed by the loop: 

j [VHjdS = J jdS 

Transforming the left-hand side according to Stokes’s theorem 
we get the circulation of the vector H over loop T: 


<J> Hdl= j jdS = J 


(/ is the current charging the capacitor). After performing similar 
calculations for surface S 2 that does not intersect the current-carrying 
wire (see Fig. 9.1), we arrive at the obviously incorrect relation 


§ Hdl= j jdS = 0 


The result we have obtained indicates that for time-varying fields 
Eq. (7.9) stops being correct. The conclusion suggests itself that this 
equation lacks an addend depending on the time derivatives of the 
fields. For stationary fields, this addend Vanishes. 

That Eq. (7.9) is not correct for non-stationary fields is also indi- 
cated by the following reasoning. Let us take the divergence of both 
sides of Eq. (7.9): 

V IVH] = Vj 

The divergence of a curl must equal zero [see Eq. (1.1061. We thus 
arrive at the conclusion that the divergence of the vector j must 
also always equal zero. But this conclusion contradicts the conti- 
nuity equation (5.11). Indeed, in non-stationary processes, p may 
change with time (this, in particular, is what happens with the charge 
density on the plates of a capacitor being charged). In this case in 
accordance with Eq. (5.11), the divergence of j differs from zero. 

To bring Eqs. (7.9) and (5.11) into agreement. Maxwell intro- 
duced an additional addend into the right-hand side of Eq. (7.9). 
It is quite natural that this addend should have the dimension of 
current density. Maxwell called it the de nsity of the displacement 
current. Thus, according to Maxwell, Eq. (7.9) should have the 
form 

[VH] = i + j d (9-8) 

The sum of the conduction current and the displacement current 
is usually called the total current. The density of the total current is 

itot = i + la 19.91 


204 


Electricity and Magnetism 


If we assume that the divergence of the displacement current equals 
that of the conduction current taken with the opposite sign: 

Vid = — Vj (9.10) 

then the divergence of the right-hand- side of Eq. (9.8), like that 
of the left-hand side, will always be zero. 

Substituting dp/dt for Vj in Eq. (9.10) in accordance with Eq. (5.11) 
we get the following expression for the divergence of the displace- 
ment current: 

Via— g- (9.11) 

To associate the displacement current with quantities characterizing 
the change in an electric field with time, let us use Eq. (2.23) accord- 
ing to which the divergence of the electric displacement vector 
equals the density of the extraneous charges: 

VD = p 

Time differentiation of this equation yields 

dt vv * dt 

Now let us change the sequence of differentiation with respect to 
time and to the coordinates in the left-hand side. As a result, we get 
the following expression for the derivative of p with respect to t: 



Introduction of this expression into Eq. (9.11) yields 

Vfe-v(f-) 

Hence, 

= (9-12) 

Using Eq. (9.12) in Eq. (9.8), we arrive at the equation 

lVH] = j + -^- (9.13) 

which, like Eq. (9.5), is one of the fundamental equations in Max- 
well’s theory. 

We must underline the fact that the term “displacement current” 
is purely conventional. In essence, the displacement current is 
a time- varying electric field. The only reason for calling the quantity 
given by Eq. (9.12) a “current” is that the dimension of this quantity 
coincides with that of current density. Of all the physical properties 


Maxwell's Equations 


205 


belonging to a real current, a displacement current has only one — 
the ability of producing a magnetic field. 

The introduction of the displacement current determined by 
Eq. (9.12) has “given equal rights” to an electric field and a magnetic 
field. It can be seen from the phenomenon of electromagnetic induc- 
tion that a varying magnetic field sets up an electric field. It follows 
from Eq. (9.13) that a varying electric field sets up a magnetic field. 

There is a displacement current wherever there is a time-varying 
electric field. In particular, it also exists inside conductors carrying 
an alternating electric current. The displacement current inside 
conductors, however, is usually negligibly small in comparison with 
the conduction current. 

We must note that Eq. (9.6) is approximate. For it to become 
quite strict, we must add a term to its right-hand side that takes 
account of the displacement current due to the weak dispersed elec- 
tric field in the vicinity of surface S v 

Let us convince ourselves that the surface integral of the right- 
hand side of Eq. (9.8) has the same value for surfaces S t and S 2 
(see Fig. 9.1). Both the conduction current and the displacement cur- 
rent due to the electric field outside the capacitor “flow” through 
surface S v Hence, for the first surface, we have 

Int,= J jdS + -£- f DdS = / + -£-<D ltm 

Si Si 

(we have changed the sequence of the operations of differentiation 
with respect to time and integration over the coordinates in the 
second addend). The quantity designated by the letter / is the current 
flowing in the conductor to the left-hand plate of the capacitor, 
in is the flux of the vector D flowing into the volume V bounded 
by surfaces and S 2 (see Fig. 9.1). 

For the second surface, j = 0, consequently 

I n t 2 = -jf j = — 0 2> 0 ut 

s* 

where 0 ut I s the flux of the vector D flowing out of volume V 
through surface S 2 . 

The difference between the integrals is 

Int 2 Int* — -jj- 0 2 , out ^ 

The current / can be represented as dq/dt , where q is the charge on 
a capacitor plate. The flux passing inward through surface equals 
the flux passing outward through the same surface taken with the 
opposite sign. Substituting — <D lf Q ut for <Di. in and dq/dt for /, 


206 


Electricity and Magnetism 


we get 

Int 2 — Int| = (^ 2 , out + <D| t out) — (®d — ?) (9-14) 

where is the flux of the vector D through the closed surface 
formed by surfaces S x and £ 2 . According to Eq. (2.25), this flux 
must equal the charge enclosed by the surface. In the given case, 
it is the charge q on a capacitor plate. Thus, the right-hand side of 
Eq. (9.14) equals zero. It follows that the magnitude of the surface 
integral of the total current density vector does not depend on the 
choice of the surface over which the integral is being calculated. 

We can construct current lines for the displacement current like 
those for the conduction current. According to Eq. (2.35), the elec- 
tric displacement in the space between the capacitor plates equals 
the surface charge density on a plate: D = a. Hence, 

• » 

D = a 

The left-hand side gives the density of the displacement current 
in the space between the plates, and the right-hand side — the density 
of the conduction current inside the plates. The equality of these 
densities signifies that the lines of the conduction current uninter- 
ruptedly transform into lines of the displacement current at the 
boundary of the plates. Consequently, the lines of the total current 
are closed. 


9.3. Maxwell’s Equations 

The discovery of the displacement current permitted Maxwell to 
present a single general theory of electrical and magnetic phenom- 
ena. This theory explained all the experimental facts known at 
that time and predicted a number of new phenomena whose existence 
was confirmed later on. The main corollary of Maxwell’s theory was 
the conclusion on the existence of electromagnetic waves propagating 
with the speed of light. Theoretical investigation of the properties 
of these waves led Maxwell to the electromagnetic theory of light. 

The theory is based on Maxwell’s equations. These equations play 
the same part in the science of electromagnetism as Newton’s laws 
do in mechanics, or the fundamental laws in thermodynamics. 

The first pair of Maxwell’s equations is formed by Eqs. (9.5) and 
(7.3): 

fVE]=— £ 

VB = 0 


(9.5) 

(7.3) 



MaxwtlYs Equations 


207 


The first of them relates the values of E to changes in the vector B 
in time and is in essence an expression of the l aw of electromagnetic 
inHucticm* The second one points to the absence of sources of a mag- 
neticTield, i.e. magnetic charges. 

The second pair of Maxwell's equations is formed by Eqs. (9.13) 
and (2.23): 

[V H ] = j + -^- (9.13) 

VD = p (2.23) 


The first of them establishes a relation between the conduction and ' 
displacement currents and the magnetic field they produce. The 
second one shows that extraneous charges are the s ourc es of the vec- 1 
tor D. 

Equations (9.5), (7.3), (9.13), and (2.23) are Maxwell's equations 
in the differential form. We must note that the first pair of equations 
includes only the basic characteristics of a field, namely, E and B. 
The second pair includes only the auxiliary quantities D and H. 

Each of the vector equations (9.5) and (9.13) is equivalent to three 
scalar equations relating the components of the vectors in the left- 
hand and right-hand sides of the equations. Using Eqs. (1.81) and 
(1.92)-(1.94), let us present Maxwell’s equation in the scalar form: 


dE z 

dEy 

K 

* 

dy 

dz 

dt 

d£x _ 

dE z _ 

dB y 

'dz 

dx 

dt 

dEy 

<a> 

H 

dB z 

dx 

dy 

dt 

dB x 


dB z 

A- 


(9.15) 


(9.16) 


(the first pair of equations), 


dff z 

% 

1 

i 4- 

dD x 

dy 

dz 

f x J 

dt 

dH x 

dH z _ 

7 —4— 

dDy 

dz 

dx 

hr 

dt 

dH v 

dH x _ 

i 4- 

dD z 

dx 

dy ~ 

Jz\ 

dt 

dD x 

l "Vh 

dP 2 

L A 

dx 

' dy ^ 

dz 

— p 


(the second pair of equations). 


(9.17) 


(9.18) 



208 


Electricity and Magnetism 


We get a total of 8 equations including 12 functions (three com- 
ponents each of the vectors E, B, D, H). Since the number of equa- 
tions is less than the number of unknown functions, Eqs. (9.5), 
(7.3), (9.13), and (2.23) are not sufficient for finding the fields accord- 
ing to the given distribution of the charges and currents. To cal- 
culate the fields, we must add equations relating D and j to E, 
and also H to B to these equations. They have the form 


D = e 0 eE 

B = |i 0 |iH 
j = cjE 


( 2 . 21 ) 
, - ( 7 - 17 ) 
cS*'- A (5.22) 


The collection of equations (9.5), (7.3), (9.13), (2.23), and (2.21), 
(7.17), (5.22) forms the foundation of the electrodynamics of media 
at rest. 

The equations 


§Eil 

BdS 

(9.19) 

r s 



§BdS = 

0 

(9.20) 

s 



(the first pair) and 



&HdI = f j dS + 

irj 0 *® 

(9.21) 

r s 


<&DdS = \ 

p dV 

(9.22) 


(the second pair) are Maxwell's equations in the integral form. 

Equation (9.19) is obtained by integration of Eq. (9.5) over arbit- 
rary surface S with the following transformation of the left-hand 
side according to Stokes’s theorem into an integral over loop T enclos- 
ing surface S. Equation (9.21) is obtained in the same way from 
Eq. (9.13). Equations (9.20) and (9.22) are obtained from Eqs. (7.3) 
and (2.23) by integration over the arbitrary volume V with the 
following transformation of the left-hand side according to the Ostro- 
gradsky-Gauss theorem into an integral over closed surface S enclos- 
ing volume V. ^ j g f 


CHAPTER 10 MOTION OF CHARGED 

PARTICLES IN ELECTRIC 
AND MAGNETIC FIELDS 

10.1. Motion of a Charged Particle in a 
Homogeneous Magnetic Field 

Imagine a charge e' moving in a homogeneous magnetic field 
with the velocity v perpendicular to B. The magnetic force imparts 
to the charge an acceleration perpendicular to the velocity 

a D =—= — vB (10.1) 

[see Eq. (6.33); the angle between v and B is a right one]. This accel- 
eration changes only the direction of the velocity, while the magni- 
tude of the latter remains unchanged. Hence, the acceleration given 
by Eq. (10.1) will be constant in magnitude too. In these conditions, 
the charged particle moves uniformly around a circle whose radius 
is determined by means of the equation a n = v*/R. Substituting 
for a n in this equation its value from Eq. (10.1) and solving the result- 
ing equation relative to i?, we get 

( 10 - 2 ) 

Thus, when a charged particle moves in a homogeneous magnetic 
field perpendicular to the plane in which the motion is taking place, 
the trajectory of the particle is a circle. The radius of the circle 
depends on the velocity of the particle, the magnetic induction of 
the field, and the ratio of the charge of the particle e' to its mass m. 
The ratio elm is called the specific charge. 

Let us find the time T needed for the particle to complete one revo- 
lution. For this purpose, we shall divide the length of the circum- 
ference 2nR by the velocity of the particle u. The result is 

T = 2n-^--j r (10.3) 

Inspection of Eq. (10.3) shows that the period of revolution of the 
particle does not depend on its velocity. It is determined only by 
the specific charge of the particle and the magnetic induction of the 
field. 

Let us determine the nature of motion of a charged particle when 
its velocity makes the angle a with the direction of a homogeneous 
magnetic field, and a is not a right angle. We shall resolve the vector 
v into two components: v ± perpendicular to B, and v§ parallel to 



210 


Electricity and Magnetism 


B (Fig. 10.1). The magnitudes of these components are 
v ± =v sin a, =v cos a 
The magnetic force has the magnitude 
F = e'vB sin a =-* e'v ± B 

and is in a plane at right angles to B. The acceleration produced by 
this force is normal for the component y^. The component of the 
magnetic force in the direction of B is zero. Hence, this force cannot 
affect the magnitude of v fl . The motion of the particle can thus be 
considered as the superposition of two motions: (1) translation along 
the direction of B with a constant velocity t; # = i; cos a, and (2) 
uniform circular motion in a plane at right angles to the vector B. 



Fig. 10.1 



Fig. 10.2 


The radius of the circle is determined by Eq. (10.2) with v x = 
=i> sin a substituted for v. The trajectory of motion is a helix (spiral) 
whose axis coincides with the direction of B (Fig. 10.2). The pitch 
of the helix l can be found by multiplying by the period of revo- 
lution T determined by Eq. (10.3): 

l as v u T = 2ji v cos a (10.4) 

The direction in which the helix curls depends on the sign of the 
particle’s charge. If the latter is positive, the helix curls counter- 
clockwise. A helix along which a negatively charged particle is 
moving curls clockwise (it is assumed that we are looking at the 
helix along the direction of B; the particle flies away from us if 
a < ji/2, and toward us if a >ji/2). 


10.2. Deflection of Moving Charged Particles 
by an Electric and a Magnetic Field 

Let us consider a narrow beam of identically charged particles 
(for example, electrons) that in the absence of fields falls on a screen 
perpendicular to it at point O (Fig. 10.3). Let us find the displace- 


Motion of Charged Particles 


211 


ment of the trace of the beam produced by a homogeneous electric 
field perpendicular to the beam and acting on a path of length l x . 
Let the initial velocity of the particles be v 0 . Upon entering the 
region of the field, each particle will move with an acceleration 
a± = (e'lm) E constant in magnitude and in direction and perpen- 
dicular to v 0 (here elm is the specific charge of a particle). Motion 



• r ,T — 3 — * 


n , 

1 LjJJ-H 



k 


! 


i 


y 


Fig. 10.3 


under the action of the field continues during the time 
During this time, the particles will be displaced over the distance 

00.5) 

and will acquire the following velocity component perpendicular 
to v 0 : 

v x = a ± t=— E 

The particles now fly in a straight line in a direction that makes 
with the vector v 0 the angle a determined by the expression 

tan a = -Z±-=—E\ (10.6) 

vq m vf v ' 

As a result in addition to the displacement given by Eq. (10.5), 
the beam receives the displacement 

y 2 = l 2 tan a — — E 

where l 2 is the distance to the screen from the boundary of the region 
which the field is in. 

The displacement of the trace of the beam relative to point 0 
is thus 

v=yi+y 2 =-^r E i$(t 1 * +12 ) 


(10.7) 



212 


Electricity and Magnetism 


Taking into account Eq. (10.6), the expression for the displacement 
can be written in the form 

y = ( y + k ) tan « 

It thus follows that the particles leaving the field fly as if they 
were leaving the centre of the capacitor setting up the field at the 
angle a determined by means of Eq. (10.6). 

Now let us assume that on a particle path of l x a homogeneous 
magnetic field is switched on perpendicular to the velocity v 0 of the 
particles (Fig. 10.4; the field is perpendicular to the plane of the 



Fig. 10.4 


drawing, the region of the field is surrounded by a dash circle). Under 
the action of the field, each particle receives the acceleration a ± = 
= (e'/m) v q B constant in magnitude. Limiting ourselves to the case 
when the deflection of the beam by the field is not great, we can consid- 
er that the acceleration a x is constant in magnitude and perpen- 
dicular to v 0 . Hence, we can use the equations we have obtained for 
calculating the displacement, replacing the acceleration a x = (e r /m)E 
in them with the value a ± = (e' hri) v 0 B. As a result, we get the fol- 
lowing expression for the displacement, which we shall now denote 
by z: 

( 10 . 8 ) 




The angle through which the beam is deflected by the magnetic 
field is determined by the expression 


tan P> = — B — 

r m u 0 

I), we can write ! 

x== (t **+*•) tan P 


(10.9) 


m uq 

With a view to Eq. (10.9), we can write Eq. (10.8) in the form 



Motion of Charged Particles 


213 


Consequently, upon small deflections, the particles after leaving 
the magnetic field fly as if they had left the centre of the region con- 
taining the deflecting field at the angle p whose magnitude is deter- 
mined by Eq. (10.9). 

Inspection of Eqs. (10.7) and (10.8) shows that both the deflection 
by an electric field and the deflection by a magnetic one are propor- 
tional to the specific charge of the particles. 

The deflection of a beam of electrons by an electric or magnetic 
field is used in cathode-ray tubes. A tube with electrical deflection 



Fig. 10.5 


(Fig. 10.5), apart from the so-called electron gun producing a narrow 
beam of fast electrons (an electron beam), contains two pairs of 
mutually perpendicular deflecting plates. By feeding a voltage to 
any pair of plates, we can produce a proportional displacement of 
the electron beam in a direction normal to the given plates. The screen 
of the tube is coated with a fluorescent composition. Therefore, 
a brightly luminescent spot appears on the screen where the electron 
beam falls on it. 

Cathode-ray tubes are used in oscillographs — instruments making 
it possible to study rapid processes. A voltage changing linearly with 
time (the scanning voltage) is fed to one pair of deflecting plates, 
and the voltage being studied to the other. Owing to the negligibly 
small inertia of an electron beam, its deflection without virtually 
any delay follows the changes in the voltages across both pairs of 
deflecting plates, and the beam draws on the oscillograph screen 
a plot of time dependence of the voltage being studied. Many non- 
electrical quantities can be transformed into electric voltages with 
the aid of the relevant devices (transducers). Consequently, oscillo- 
graphs are used to study the most diverse processes. 

A cathode-ray tube is an integral part of television equipment. 
In television, tubes with magnetic control of the electron beam are 
used most frequently. In these tubes, the deflecting plates are replaced 
with two external mutually perpendicular systems of coils each 
of which sets up a magnetic field perpendicular to the beam. Chang- 
ing of the current in the coils produces motion of the light spot created 
by the electron beam on the screen. 


10.3. Determination of the Charge and Mass 
of an Electron 

The specific charge of an electron (i.e. the ratio elm) was first 
measured by the British physicist Joseph J. Thomson (1856-1940) in 
1897 with the aid of a discharge tube like the one shown in Fig. 10.6. 
The electron beam (cathode rays; see Sec. 12.6) emerging from the 
opening in anode A passed between the plates of a parallel-plate 
capacitor and impinged on a fluorescent screen producing a light 
spot on it. By feeding a voltage to the capacitor plates, it was pos- 
sible to act on the beam with a virtually homogeneous electric field. 



Fig. 10.6 


The tube was placed between the poles of an electromagnet, which 
could produce a homogeneous magnetic field perpendicular to the 
electric one on the same portion of the path of the electrons (the region 
of the magnetic field is shown in Fig. 10.6 by the dash circle). When 
the fields were switched off, the beam impinged on the screen at 
point O. Each of the fields separately caused deflection of the beam 
in a vertical direction. The magnitudes of the displacements were 
determined with the aid of Eqs. (10.7) and (10.8) obtained in the 
preceding section. 

After switching on the magnetic field and measuring the displace- 
ment of the beam trace 

( 101 °) 

which it produced, Thomson also switched on the electric field 
and selected its value so that the beam would again reach point O. 
In this case, the electric and magnetic fields acted on the electrons 
of the beam simultaneously with forces identical in value, but 
opposite in direction. The condition was observed that 

eE = ev 0 B (10.11) 

By solving the simultaneous equations (10.10) and (10.11), Thomson 
calculated e/m and v 0 . 

H. Busch used the method of magnetic focussing to determine the 
specific charge of electrons. The essence of this method consists in 


Motion of Charged Particles 


215 


the following* Assume that a slightly diverging beam of electrons 
having a velocity v identical in magnitude flies out from a certain 
point of a homogeneous magnetic field. The beam is symmetrical 
relative to the direction of the field. The directions in which the elec- 
trons fly out form small angles a with the direction of B. It was shown 
in Sec. 10.1 that the electrons in this case travel along helical trajec- 
tories, performing during the identical time 

T = 2n 

a complete revolution and being displaced along the direction of the 
held over the distance l equal to 

l — v cos a*T (10.12) 

Owing to the smallness of the angles a, the distances (10.12) for 
different electrons are virtually the same and equal vT (for small 



Fig. 10.7 


angles cos a « 1). Consequently, the slightly diverging beam is 
focussed at a point that is at the distance 

l = vT = 2n — 

e B 

from the point of emergence of the electrons. 

In Busch’s experiment, the electrons emitted by hot cathode C 
(Fig. 10.7) are accelerated when passing through the potential differ- 
ence U applied between the cathode and anode A. As a result, they 
acquire the velocity v whose value can be found from the relation 

(10.14) 

After next flying out through an opening in the anode, the electrons 
form a narrow beam directed along the axis of the evacuated tube 
inserted into a solenoid. A capacitor fed with a varying voltage is 
placed at the inlet of the solenoid. The field set up by the capacitor 
deflects the electrons of the beam from the axis of the instrument 
through small angles a changing with time. This leads to “eddying” 
of the beam — the electrons begin to move along different helical 
trajectories. A fluorescent screen is placed at the outlet from the sole- 
noid. If the magnetic induction B is selected so that the distance V 



(10.13) 


216 


Electricity and Magnetism 


from the capacitor to the screen complies with the condition 

V « nl (10.15) 

(Z is the pitch of the helix, and n is an integer), then the point of 
intersection of the trajectories of the electrons gets onto the screen — 
the electron beam is focussed at this point and produces a sharp 
luminescent spot on the screen. If condition (10.15) is not observed, 
the luminescent spot on the screen will be blurred. We can find 
elm and v by solving the system of equations (10.13), (10.14), and 
(10.15). 

The most accurate value of the specific charge of an electron estab- 
lished with account taken of the results obtained by different 
methods, is 

-£-=1.76 x 10 11 C/kg = 5.27 x 10” cgse,/g (10.16) 


Equation (10.16) gives the ratio of the charge of an electron to its 
rest mass m. In the experiments conducted by Thomson, Busch, 

and in other similar experiments, the 
ratio of the charge to the relativistic 
mass 



m T = — 7 ===- (10.17) 

/l — V */ C t V ' 

was determined. In Thomson’s experi- 
ments, the speed of the electrons was 
about 0.1c. At such a speed, the rela- 
tivistic mass exceeds the rest mass by 
Fig. 10.8 0.5%. In subsequent experiments, the 

speed of the electrons reached very high 
values. In all cases, the experimenters discovered a reduction in the 
measured values of elm with a growth in v, which occurred in com- 
plete accordance with Eq. (10.17). 

The charge of an electron was determined with high accuracy by 
the American scientist Robert Millikan (1886-1953) in 1909. He 
introduced very minute oil droplets into the closed space between 
horizontally arranged capacitor plates (Fig. 10.8). When atomized, 
the droplets became electrolyzed, and they could be suspended in 
mid air by properly choosing the magnitude and the sign of the volt- 
age across the capacitor. Equilibrium set in when the following 
condition was observed: 

P' = e'E (10.18) 

Here e f is the charge of a droplet, and P* is the resultant of the force 
of gravity and the buoyant force equal to 

P'=*^nr 3(p — p 0 )g 


(10.19) 



Motion of Charged Particles 217 

(p is the density oi a droplet, r is its radius, and p 0 is the density 
of air). 

Equations (10.18) and (10.19) can be used to find e if we know r. 
To determine the radius, the speed v 0 of uniform falling of a droplet 
was measured in the absence of a field. Uniform motion of a droplet 
sets in provided that the force P' is balanced by the force of resist- 
ance F = 6m|n> (see Eq. (9.24) of Vol. I, p. 264; tj is the viscosity 
of air]: 

P' — 6nt)n; 0 (10.20) 

The motion of a droplet was observed with the aid of a microscope. 
To measure v 0 , the time was determined during which a droplet 
covered the distance between two threads that could be seen in the 
field of vision of the microscope. 

It is very difficult to accurately suspend a droplet in equilibrium. 
Therefore, instead of a field complying with condition (10.18), 
such a field was switched on under whose action a droplet began 
to move upward with a small speed. The steady speed of rising v s 
is determined from the condition that the force P' and the force 
6nt)n; together balance the force e'E: 

P' + 6nt|n;* = e'E (10. 21) 

Excluding P' and r from Eqs. (10.19), (10.20), and (10.21), we 
get an expression for e' : 

e ' = 9n < 10 - 22 > 

(Millikan introduced a correction into this equation taking into 
account that the dimensions of the droplets were comparable with 
the free path of air molecules). 

Thus, by measuring the speed of free fall of a droplet v Q and the 
speed of its rise v E in a known electric field E y one could find the 
charge of a droplet e'. In measuring the speed v B at a certain value 
of the charge e\ Millikan ionized the air by radiating X-rays through 
the space between the plates. Separate ions adhered to a droplet and 
changed its charge. As au result, the speed v B also changed. After 
measuring the new value of the speed, the space between the plates 
was again irradiated, and so on. 

The changes in the charge of a droplet Ae' and the charge e ' itself 
measured by Millikan were each time found to be integral multiples 
of the same quantity e. This was an experimental proof of the discrete 
nature of an electric charge, i.e. of the fact that any charge consists 
of elementary charges of the same magnitude. 


218 


Electricity and Magnetism 


The value of the elementary charge established with a view to 
Millikan’s measurements and to the data obtained in other ways is 

1.60 x 10 -1 * C = 4.80 x 10“ 10 cgse q (10.23) 

The charge of an electron has the same value. 

The rest mass of an electron obtained from Eqs. (10.16) and (10.23) 


is 


m = 0.91 x 10- 30 kg = 0.91 x 10“ 27 g 


(10.24) 

It is about 1/1840 of the mass of the lightest of all atoms — the hydro- 
gen atom. 

The laws of electrolysis established experimentally by Michael 
Faraday in 1836 played a great part in discovering the discrete nature 
of electricity. According to these laws, the mass m of a substance 
liberated when a current passes through an electrolyte* is proportion- 
al to the charge q carried by the current: 

1 


m — — q 

F z * 


(10.25) 


Here M — mass of one mole of the liberated substance 
% — valence of the substance 

F = Faraday’s constant (Faraday’s number) equal to 

F = 96.5 X 10 3 C/mol (10.26) 

Dividing both sides of Eq. (10.25) by the mass of an ion, we get 

where N ± = Avogadro’s constant 

N = number of ibns contained in the mass to. 

Hence, for the charge of one ion, we have 

F 


*' = JL 
* TV 




Consequently, the charge of an ion is an integral multiple of the 
quantity 



(10.27) 


which is the elementary charge. 

Thus, the discrete nature of the charges which ions in electrolytes 
can have follows from an analysis of the laws of electrolysis. 

* Electrolytes are solutions of salts, alkalies or acids in water and some other 
liquids, and also molten salts that are ionic crystals in the solid state. Chemical 
transformations occur in electrolytes when a current passes through them. Such 
substances are called electrolytic conductors (conductors of the second kind) 
to distinguish them from electronic conductors (conductors of the first kind) 
in which the passage of a current is not attended by chemical transformations. 


Motion of Charged Particles 


219 


Substituting for F in Eq. (10.27) its value from Eq. (10.26) and 
for N a its value found from J. Perrin’s experiments (see Sec. 11.9 
of Vol. I, p. 330), we get a value for e that agrees quite well with that 
found by Millikan. 

Since the accuracy with which Faraday’s constant is determined 
and the accuracy of the value of e obtained by Millikan are greatly 
superior to the accuracy of Perrin’s experiments for determining N A , 
Eq. (10.27) was used to determine Avogadro’s constant. Here the 
value of F found from experiments in electrolysis and the value of 
e obtained by Millikan were used. 


10.4. Determination of the Specific Charge 
of Ions. Mass Spectrographs 

The methods of determining the specific charge described in the 
preceding section are suitable when all the particles in a beam have 
the same velocity. All the electrons forming a beam are accelerated 
by the same potential difference 
applied between the cathode from 
which they fly out and the anode. 

Therefore, the scattering of the va- 
lues of the velocities of the electrons 
in a beam is very small. If matters 
were different, an electron beam 
would produce a greatly blurred spot 
on the screen, and measurements 
would be impossible. 

Ions are formed as a result of ioni- 
zation of molecules of a gas that 
takes place in a volume having an 
appreciable length. Appearing in 
different places of this volume, the 
ions then pass through different 
potential differences, and, consequently, their velocities are different. 
Thus, the methods used to determine the specific charge of electrons 
cannot be applied to ions. In 1907, J. J. Thomson developed the 
“method of parabolas”, which made it possible to circumvent the 
difficulty noted above. 

In Thomson’s experiment, a narrow beam of positive ions passed 
through a region in which it simultaneously experienced the action 
of parallel electric and magnetic fields (Fig. 10.9). Both fields were 
virtually homogeneous and made a right angle with the initial di- 
rection of the beam. They produced deflections of the ions: the magnet- 
ic field deflected them in the direction of the x-axis, the electric one 




220 


Electricity and Magnetism 


along the y -axis. According to Eqs. (10.8) and (10.7), these deflections 
are 


m v 

y = — E\ 




(10.28) 


where v = velocity of a given ion with the specific charge e'/m 

= length of the region in which the field acts on the beam 
l t — distance from the boundary of this region to the photo- 
graphic plate registering the ions impinging on it. 
Equations (10.28) are the coordinates of the point at which an ion 
having the given values of e'/m and the velocity v impinges on the 



plate. Ions having the same specific charge, but different velocities, 
reached different points of the plate. Eliminating the velocity v 
from Eqs. (10.28), we get the equation of a curve along which the 
traces of ions having the same value of e'/m are arranged: 


E m 2 

y — ~s x 


(10.29) 


Inspection o! Eq. (10.29) shows that ions having identical values 
of e'/m and different values of v left a trace in the form of a parabola 
on the plate. Ions having different values of e'/m occupied different 
parabolas. Equation (10.29) can be used to find the specific charge of 
the ions corresponding to each parabola if the parameters of the in- 
strument are known (i.e. E , B , Z 1? and Z 2 ), and the displacements x 
and y are measured. When the direction of one of the fields was re- 
versed, the relevant coordinate reversed its sign, and parabolas sym- 
metrical to the initial ones were obtained. Dividing the distance be- 
tween similar points of symmetrical parabolas in half made it possi- 




Motion of Charged Particles 


221 


ble to find x and y . The trace left on the plate by the beam with the 
fields switched off gave the origin of coordinates. Figure 10.10 shows 
the first parabolas obtained by Thomson. 

When performing experiments with chemically pure neon, Thom- 
son discovered that this gas produced two parabolas corresponding 
to relative atomic masses of 20 and 22. This result gave rise to the 
assumption that there are two chemically indistinguishable varieties 
of the neon atoms (today we call them isotopes of neon). This assump- 



tion was proved by the British scientist Francis Aston (1877-1945), 
who improved the method of determining the specific charge of ions. 

Aston’s instrument, which he called a mass spectrograph, was de- 
signed as follows (Fig. 10.11). A beam of ions separated by a system 
of slits was consecutively passed through an electric field and a mag- 
netic field. These fields were directed so that they caused the ions to 
travel to opposite sides. When they passed through the electric field, 
ions with a given value of e' I m were deflected more when their veloc- 
ity was lower. Consequently, the ions left the electric field in the 
form of a diverging beam. The trajectories of the ions were also curved 
more in the magnetic field when their velocity was lower. Since 
the ions were deflected to opposite sides by the two fields, after leav- 
ing the magnetic field they formed a beam converging at one point. 

Ions with other values of the specific charge were focussed at other 
points (the trajectories of the ions for only one value of e' !m are 
shown in Fig. 10.11). The relevant calculations show that points at 
which beams formed by ions having different values of e' /m converge 
are approximately on a single straight line (shown by a dash line in 
the figure). Putting a photographic plate along this line, Aston 
obtained a number of short lines on it, each of which corresponded 
to a definite value of e' !m. The similarity of the image obtained 
on the plate to a photograph of an optical line spectrum was the 
reason why Aston called it a mass spectrogram, and the instrument 
itself— a mass spectrograph. Figure 10.12 shows mass spectrograms 
obtained by Aston (the mass numbers of the relevant ions are indi- 
cated opposite the lines). 



222 


Electricity and Magnetism 


K. Bainbridge designed an instrument of a different kind. In the 
Bainbridge mass spectrograph (Fig. 10.13), a beam of ions first passes 
through the so-called velocity selector that separates ions having a 
definite velocity from the beam. In the selector, the ions experience 
the action of mutually perpendicular electric and magnetic fields 



Fig. 10.12 


that deflect the ions to opposite sides. Only those ions pass through 
the selector slit for which the actions of the electric and magnetic 
fields compensate each other. This occurs when e'E = e'vB. Hence, 
the velocities of the ions leaving the selector regardless of their mass 

and charge have identical values 
equal to v = E/B. 

After leaving the selector, the ions 
get into the region of a homogeneous 
magnetic field of induction B' at 
right angles to their velocity. In this 
field, they move along circles whose 
radii depend on e'/m: 

u m v 

[see Eq. (10.2)1. After completing 
a semi-circle, the ions strike a pho- 
tographic plate at distances of 2 R 
from the slit. Hence, the ions of each 
species (determined by the value of e’/m) leave a trace on the plate 
in the form of a narrow strip. The specific charges of the ions can 
be calculated if the parameters of the instrument are known. Since 
the charges of the ions are integral multiples of the elementary 
charge e, the. masses of the ions can be calculated from the found va- 
lues of e'lm. 





Motion of Charged Particles 


223 


Numerous kinds of mass spectrographs are in use at present. 
Instruments have also been designed in which the ions are registered 
by means of an electrical device instead of by a photographic plate. 
They are called mass spectrometers. 


10.5. Charged Particle Accelerators 

Experiments using beams of high-energy charged particles play 
a great part in the physics of atomic nuclei and elementary particles. 
The devices used for obtaining such beams are called charged particle 
accelerators. There are many types of such devices. We shall acquaint 
ourselves with the operating principles of 
some of them. 

The Van De Graaff Generator. In 1929, R. 
van de Graaff proposed an electrostatic gen- 
erator based on the fact that surplus charges 
take up a position on the external sur- 
face of a conductor. A schematic view of the 
generator is shown in Fig. 10.14. A hollow 
metal sphere called a conductor is mounted 
on an insulating column. An endless moving 
belt of silk or rubberized fabric mounted on 
shafts is introduced into the sphere. A comb 
of sharp points is installed at the base of the 
column near the belt. The charge produced 
by a voltage generator (VG) for several 
scores of kilovolts flows onto the belt from 
the comb points. The conductor contains a 
second comb onto whose points the charge 
flows from the belt. This comb is connected 
to the conductor so that the charge taken off the belt immediately pass* 
es over to its external surface. As charges accumulate on the conduc- 
tor, its potential grows until the charge that leaks away becomes equal 
to the newly supplied charge. The leakage is mainly due to ioniza- 
tion of the gas near the surface of the conductor. The resulting passage 
of a current through the gas is called a corona discharge (see 
Sec. 12.8). The surface of the conductor is carefully polished to reduce 
the corona discharge. 

The potential up to which the conductor can be discharged is 
limited by the circumstance that at a field strength of about 3 MV/m 
(30 kV/cm) a discharge appears in the air at atmospheric pressure. 
For a sphere, E = <p/r. Therefore, to obtain higher potential differ- 
ences, the size of the conductor has to be increased (up to 10 m in 
diameter). The maximum potential difference that can be obtained 




224 


Electricity and Magnetism 


in practice with the aid of a van de Graaff generator is about 10 MV 
(10 7 V). 

Particles are accelerated in a discharge tube ( DT ) to whose elec- 
trodes the potential difference obtained in the generator is applied. 
A van de Graaff generator is sometimes designed in the form of two 
identical columns near each other whose conductors are charged op- 
positely. In this case, the discharge tube is connected between the 
conductors. 

It must be noted that the generator belt, conductor, discharge 
tube, and the earth form a closed direct current circuit. Inside the 

tube, the charges move under the action 
of the electrostatic field. Charges are car- 
ried to the conductor from the earth by 
extraneous forces whose part is played by 
the mechanical forces bringing the gen- 
erator belt into motion. 

Betatron. This is the name given to an 
induction accelerator of electrons using 
a vortex electric field. It consists of a to- 
roidal evacuated chamber (a doughnut) 
placed between the poles of an electromagnet of a special shape (Fig. 
10.15). The winding of the magnet is supplied with alternating cur- 
rent having a frequency of about 100 Hz. The varying magnetic field 
produced performs two functions: first, it sets up a vortex electric 
field accelerating the electrons, and, second, it retains the electrons 
in an orbit coinciding with the axis of the doughnut. 

To keep an electron in an orbit of constant radius, the magnetic 
induction of the field must be increased as its velocity grows [accord- 
ing to Eq. (10.2), the radius of the orbit is proportional to v/B]. Con- 
sequently, only the second and fourth quarters of the current period 
can be used for acceleration because at their beginning the current in 
the magnet winding is zero. A betatron thus operates in pulse condi- 
tions. At the beginningof the pulse, an electron gun feeds a beam of 
electrons into the doughnut. The beam is caught up by the vortex 
electric field and begins to travel in a circular orbit with a constantly 
growing velocity. During the growth of the magnetic field (about 
10~ 3 s), the electrons are able to complete up to a million revolutions 
and acquire an energy that may reach several hundred MeV. With 
such an energy, the speed of the electrons almost equals the speed of 
light c. 

For an electron being accelerated to travel in a circular orbit of 
radius r 0 , a simple relation, which we shall now proceed to derive, 
must be observed between the magnetic induction of the field in the 
orbit and inside it. The vortex electric field is directed along a tan- 
gent to the orbit along which the electron is travelling. Hence, the 
rirmiatinn of tho vp.r.tnr F » loner this orbit is 2nr n E. At the same 



|M I 


I. ll 
Fig. 10.15 



mitt re dm 





Motion of Charged Particles 


225 


time according to Gq. (9.19), the circulation of the vector E 
is — ( d<t>/dt ), where <t> is the magnetic flux through the surface en- 
closed by the orbit. The minus sign indicates the direction of E. 
We shall be interested only in the magnitude of the field strength, 
therefore we shall omit the minus sign. Equating the two expres- 
sions for the circulation, we find that 

p__ 1 d<S> 

L ~ 2 nr 0 ~dt 

The magnetic field is perpendicular to the plane of the orbit. We 
can therefore assume that O = nrj {B), where (B) is the average 
value of the magnetic induction over the area of the orbit. Hence, 

Let us write the relativistic equation of motion of an electron in 
orbit: 

£( ~»-v ) =eE+ ‘ i,B -’ rt| <io - si) 

(B 0 rb Is the magnetic induction of the field in the orbit). 

The velocity of an electron moving along a circle of radius r 0 can 
be written in the form v = ©r 0 x, where <■> is the angular velocity of 
the electron, and x is the unit vector of a tangent to the orbit. The 
vector E can be represented in the form 


Isee Eq. (10.30)]. Finally, the product [vB] can be written in the 
form vBn = <or 0 Z?n, where n is a unit vector of a normal to the orbit. 
In view of what has been said above, let us write Eq. (10.31) as follows: 


d / mcorpT \ er 0 d 

dt \ Y \ ' 2 dt 


(B) x -f eoir^orbn 


(10.32) 


The time derivative of the unit vector t is t = con [see Eq. (1.56) 
of Vol. I, p. 35; the angular velocity of rotation of the unit vector t 
coincides with the angular velocity of an electron]. Consequently, 
performing differentiation in the left-hand side of Eq. (10.32), we 
arrive at the equation 


d i mar p \ T _| mcor 0 

dt \ 1 — ©*rj/c* / "T” Y 1— ©Vj/c* 


mn = -^2- (B) x -]- e(or 0 B orb n 


Equating the factors of similar unit vectors in the left-hand and right- 
hand sides of the equation, we get 


d_ 

dt 


morp 

y i— (ovj/c^ 

nuorp 

Y 1 — <D*rJ/c* 


)=-^4-(s> 

= er qB 0 r jj 


(10.33) 

(10.34) 


226 


Electricity and Magnetitm 


It follows from Eq. (10.33) that 


ma>ro 


__ era 


<B > 


Y 1— c o*rg/e* 

(co and (2?) at the beginning of a pulse equal zero). 

A comparison of Eqs. (10.34) and (10.35) yields: 

^orb = 4 (B) 


(10.35) 


Thus, for an electron to travel constantly in a circular orbit, the mag- 
netic induction in the orbit must be half of the average value of the 
magnetic induction inside the orbit. This is achieved by making the 

pole shoes in the form of truncated cones 
(see Fig. 10.15). 

At the end of an acceleration cycle, an 
additional magnetic field is switched on 
that deflects the accelerated electrons from 
their stationary orbit and directs them 
onto a special target inside the doughnut. 
Upon striking the target, the electrons 
emit hard electromagnetic radiation 
(gamma rays, X-rays). 

Betatrons are mainly used in nuclear 
investigations. Small accelerators for an 
energy up to 50 MeV have found use in industry as sources of very 
hard X-rays employed for flaw detection in massive articles. 

Cyclotron. The accelerator bearing this name is based on the period 
of revolution of a charged particle in a homogeneous magnetic field 
being independent of its velocity [see Eq. (10.3)]. This apparatus 
consists of two electrodes in the form of halves of a low round box 
(Fig. 10.16) called dees. The latter are confined in an evacuated hous- 
ing placed between the poles of a large electromagnet. The field pro- 
duced by the magnet is homogeneous and perpendicular to the plane 
of the dees. The dees are supplied with an alternating voltage pro- 
duced by a high-frequency generator. 

Let us introduce a charged particle into the slit between the dees 
at the moment when the voltage reaches its maximum value. The 
particle will be caught up by the electric field and pulled into one of 
the dees. The space inside the dee is equi potential, therefore the par- 
ticle in it will be under the action of only a magnetic field. In this 
case, the particle travels along a circle whose radius is proportional 
to the velocity of the particle [see Eq. (10.2)]. Let us choose the fre- 
quency of the change in the voltage between the dees so that by the 
moment when the particle, after covering half of the circle, approach- 
es the slit between the dees, the potential difference between them 
will change its sign and reach its amplitude value. The particle 
will now be accelerated again and fly into the second dee with an 



Fig. 10.16 


Motion of Charged Particles 


227 


energy double that with which it travelled in the first dee. Having 
a greater velocity, the particle will travel in the second dee along a 
circle of a greater radius ( R is proportional to p), but the time during 
which it covers half the circle remains the same as previously. There- 
fore by the moment when the particle flies into the slit between the 
dees, the voltage between them will again change its sign and take 
on the amplitude value. 

Thus, the particle travels along a curve close to a spiral, and each 
time it passes through the slit between the dees it receives an addi- 
tional portion of energy equal to e'U m (e' is the charge of the particle, 
and U m is the amplitude of the voltage produced by the generator). 
Having a source of alternating voltage of a comparatively small val- 
ue (U m is about 10 5 V) at our disposal, we can use a cyclotron to 
accelerate protons up to energies of about 25 MeV. At higher energies, 
the dependence of the mass of the protons on the velocity begins to 
tell — the period of revolution increases [according to Eq. (10.3) it 
is proportional to ml, and the synchronism between the motion of 
the particles and the changes in the accelerating field is violated. 

To prevent this violation of synchronism and to obtain particles 
having higher energies, either the frequency of the voltage fed to the 
dees or the magnetic field induction is made to vary. An apparatus in 
which in the course of accelerating each portion of particles the fre- 
quency of the accelerating voltage is diminished as required is called 
a phasotron (or a synchrocyclotron). An accelerator in which the fre- 
quency remains constant, while the magnetic field induction is 
changed so that the ratio m/B remains constant is called a synchrotron 
(equipment of this type is used only to accelerate electrons). 

In the accelerator called a synchrophasotron or a proton synchrotron, 
both the frequency of the accelerating voltage and the magnetic 
field induction are changed. The particles being accelerated travel 
in this machine along a circular path instead of a spiral. An increase 
in the velocity and mass of the particles is attended by a growth in 
the magnetic field induction so that the radius determined by 
Eq. (10.2) remains constant. The period of revolution of the particles 
changes both owing to the growth in their mass and to the growth in 
B . For the accelerating voltage to be synchronous with the motion 
of the particles, the frequency of this voltage is made to change ac- 
cording to the relevant law. A synchrophasotron has no dees, and the 
particles are accelerated on separate sections of the path by the 
electric field produced by the varying frequency voltage generator. 

The most powerful accelerator at present (in 1979) — a proton syn- 
chrotron — was started in 1974 at the Fermi National Accelerator 
Laboratory at Batavia, Illinois, in the USA. It accelerates protons 
up to an energy of 400 GeV (4 X 10 11 eV). The speed of protons hav- 
ing such an energy differs from that of light in a vacuum by less than 
0.0003% (v = 0.999 997 2c). 




CHAPTER 11 THE CLASSICAL THEORY 

OF ELECTRICAL 
CONDUCTANCE 
OF METALS 

11.1. The Nature of Current Carriers 
in Metals 

A number of experiments were run to reveal the nature of the cur- 
rent carriers in metals. Let us first of all note the experiment conduct- 
ed in 1901 by the German physicist Carl Riecke (1845-1915). He 
took three cylinders — two of copper and one of aluminium — with 
thoroughly polished ends. After being weighed, the cylinders were 
put end to end in the sequence copper-aluminium-copper. A current 
was passed in one direction through this composite conductor during 

a year. During this time, a total charge of 
3.5 X 10 6 C passed through the cylinders. 
Weighing showed that the passage of a 
current had no effect on the weight of the 
cylinders. When the ends that had been 
in contact were studied under a micro- 
scope, no penetration of one metal into 
Fig. 11.1 another was detected. The results of the 

experiment indicate that a charge is car- 
ried in metals not by atoms, but by particles encountered in all met- 
als. The electrons discovered by J. J. Thomson in 1897 could be 
such particles. 

To identify the current carriers in metals with electrons, it was 
necessary to determine the sign and numerical value of the specific 
charge of the carriers. Experiments run for this purpose were based 
on the following considerations. If metals contain charged particles 
capable of moving, then upon the deceleration (braking) of a metal 
conductor these particles should continue to move by inertia for a 
certain time, as a result of which a current pulse will appear in the 
conductor, and a certain charge will be carried in it. 

Assume that a conductor initially moves with the velocity v 0 
(Fig. 11.1). We shall begin to decelerate it with the acceleration a. 
Continuing to move by inertia, the current carriers will acquire the 
acceleration — a relative to the conductor. The same acceleration can 
be imparted to the carriers in a stationary conductor if an electric 
field of strength E = — male 1 is set up in it, i.e. the potential 
difference 2 2 

<Pi-q> 2 = ( Edi=- ( = 





Classical Theory of Electrical Conductance of Metals 


229 


is applied to the ends of the conductor (m and e ' are the mass and 
the charge of a current carrier, l is the length of a conductor). In 
this case, the current Z = (qp 1 — q> 2 )/i?, where R is the resistance of 
the conductor, will flow through it (/ is considered to be positive if 
the current flows in the direction of motion of the conductor). Hence, 
the following charge will pass through each cross section of the con- 
ductor during the time dt: 

dq = I dt= dt ~dv 

The charge passing during the entire time of deceleration is 

<«•*> 

0 90 

(the charge is positive if it is carried in the direction of motion of 
the conductor). 

Thus, by measuring /, i; 0 , and R , and also the charge q flowing 
through the circuit when the conductor is decelerated, we can find 
the specific charge of the carriers. The direction of the current pulse 
will indicate the sign of the carriers. 

The first experiment with conductors moving with acceleration 
was run in 1913 by the Soviet physicists Leonid Mandelshtam (1879- 
1944) and Nikolai Papaleksi (1880-1947). They made a wire coil 
perform rapid torsional oscillations about its axis. A telephone was 
connected to the ends of the coil, and the sound due to the current 
pulses was heard in it. 

A quantitative result was obtained by the American physicists 
R. Tolman and T. Stewart in 1916. A coil of a wire 500 m long was 
made to rotate with a linear velocity of the turns of 300 m/s. The 
coil was then sharply braked, and a ballistic galvanometer was used 
to measure the charge flowing in the circuit during the braking time. 
The value of the specific charge of the carriers calculated by Eq. 
(11.1) wasobtained very close to elm for electrons. It was thus proved 
experimentally that electrons are the current carriers in metals. 

A current can be produced in metals by an extremely small poten- 
tial difference. This gives us the grounds to consider that the current 
carriers — electrons — move without virtually any hindrance in a 
metal. The result of Tolman’s and Stewart’s experiment lead to the 
same conclusion. 

The existence of free electrons in metals can be explained by the 
fact that when a crystal lattice is formed, the most weakly bound 
(valence) electrons detach themselves from the atoms of the metal. 
They become the “collective” property of the entire piece of metal. 
If one electron becomes detached from every atom, then the concen- 
tration of the free electrons (i.e. their number n in a unit volume) 



230 


Electricity and Magnetism 


will equal the number of atoms in a unit volume. The latter number 
is ( 6/M) N A , where 6 is the density of the metal, M is the mass of 
a mole, N A is Avogadro’s constant. The values of 8/M for metals 
range from 2 X 10* mol/m 3 (for potassium) to 2 x 10 5 mol/m 3 (for 
beryllium). Hence, we get values of the following order for the con- 
centration of the free electrons (or conduction electrons, as they are 
also called): 

n = 10 28 -10 29 m' 3 (10 22 -10 23 cm -3 ) (11.2) 


11.2. The Elementary Classical Theory 
of Metals 


Proceeding from the notions of free electrons, the German physi- 
cist Paul Drude (1863-1906) created the classical theory of metals that 
was later improved by H. Lorentz. Drude assumed that the conduc- 
tion electrons in a metal behave like the molecules of an ideal gas. 
In the intervals between collisions, they move absolutely freely, 
covering on an average a certain path Z. True, unlike the molecules 
of a gas whose free path is determined by collisions of the molecules 
with one another, the electrons collide chiefly not with one another, 
but with the ions forming the crystal lattice of the metal. These col- 
lisions result in the establishment of thermal equilibrium between 
the electron gas and the crystal lattice. 

Assuming that the results of the kinetic theory of gases may be 
extended to an electron gas, we can use the following formula to as- 
sess the average velocity of thermal motion of the electrons: 


<*> 


-v- 


SkT 

Jim 


(11.3) 


[see Eq. (11.65) of Vol. I, p. 320]. Calculations by this equation for 
room temperature (about 300 K) give the following result: 


<v>=y r 


8 X 1.38 X 10-* 3 X 300 
3.14 x 0.91 xlO" 30 


10 5 m/s 


When a field is switched on, the ordered motion of the electrons 
with a certain average velocity <u> is superposed onto the chaotic 
thermal motion occurring with the velocity (i>). It is simple to assess 
the value of <u) by the equation 

/ = ne(u > (11.4) 


(see Eq. (5.23)1. The maximum current density for copper wires al- 
lowed by the relevant specifications is about 10 7 A/m 2 (10 A/mm 2 ). 
Taking the value of 10 29 m -3 for n, we get 


en 


1.6 x 10' 1 * X 10 s * 


10~ 3 m/s 


Classical Theory of Electrical Conductance of Metals 


231 


Thus, even at very high current densities, the average velocity of 
ordered motion of the charges (u) is about 1/10 8 of the average veloc- 
ity of thermal motion < v >. Therefore in calculations, the magnitude 
of the resultant velocity | v + u | may be replaced with that of the 
velocity of thermal motion | v |. 

Let us find the change in the mean value of the kinetic energy of 
the electrons produced by a field. The mean square of the resultant 
velocity is 

<(v + u) 2 > = <v 2 -f- 2vu + u 2 ) = <v 2 > + 2 <vu> -{- <u 2 ) (11.5) 

The two events consisting in that the velocity of thermal motion of 
the electrons will take on the value v, while the velocity of ordered 
motion — the value u, are statistically independent. Therefore, ac- 
cording to the theorem on the multiplication of probabilities [see 
Eq. (11.4) of Vol. I, p. 296], we have (vu> = (v) <u>. But (v> 
equals zero, so that the second addend in Eq. (11.5) vanishes, and 
it acquires the form 

<(v + u) 2 > = <v 2 > + <u 2 > 

Hence, it follows that the ordered motion increases the kinetic energy 
of the electrons on an average by 

<Ae k > = -2^- (11.6) 

Ohm’s Law. Drude considered that when an electron collides with 
an ion of the crystal lattice, the additional energy (11.6) acquired 
by the electron is transmitted to the ion and, consequently, the ve- 
locity u as a result of the collision vanishes. Let us assume that the 
field accelerating the electrons is homogeneous. Hence, under the 
action of the field, the electron receives a constant acceleration equal 
to eE/nty and toward the end of its path the velocity of ordered mo- 
tion will reach, on an average, the value 

= (11.7) 

where t is the average time elapsing between two consecutive col- 
lisions of the electron with ions of the lattice. 

Drude did not take into consideration the distribution of the elec- 
trons by velocities and ascribed the same value of the velocity v 
to all the electrons. In this approximation 



(we remind our reader that | v + u | virtually equals | v |). Using 
this value of t in Eq. (11.7), we get 


eEl 



232 


Electricity and Magnetism 


The velocity u changes linearly during the time it takes to cover 
the path 1. Therefore, its average value over the path equals half the 
maximum value: 


<u> 


l u _ fEl 
2 Umax “ 2mv 


Introducing this equation into Eq. (11.4), we get 


7 = 


neH 
2 mv 


E 


The current density is found to be proportional to the field strength. 
We have thus arrived at Ohm’s law. According to Eq. (5.22), the 
constant of proportionality between / and E is the conductivity 


a = 


neH 

2mv 


(11.9) 


If the electrons did not collide with the ions of the lattice, their free 
path and, consequently, the conductivity of the metal would be 
infinitely great. Thus, according to the classical notions, the elec- 
trical resistance of metals is due to the collisions of their free elec- 
trons with the ions at the crystal lattice points of the metal. 

The Joule- Lenz Law* By the end of its free path, an electron 
acquires additional kinetic energy whose average value is 

<Ae k > = ^| ^ = (H.10) 

[see Eqs. (11.6) and (1 x .8)1. Upon colliding with an ion, the electron, 
according to the assumption, completely transfers the additional 
energy it has acquired to the crystal lattice. The energy given up to 
the lattice goes to increase the internal energy of the metal, which 
manifests itself in its becoming heated. 

Every electron experiences on an average 1 /t = vll collisions a 
second, communicating each time the energy expressed by Eq. (11.10) 
to the lattice. Hence, the following amount of heat should be liberat- 
ed in unit volume per unit time: 


(n is the number of conduction electrons per unit volume). 

The quantity Q u is the unit thermal power of a current (see Sec. 5.8). 
The factor of E 2 coincides with the value given by Eq. (11.9) for a. 
Passing over in the expression aE 2 from a and E to p and /, we arrive 
at the formula Q u = p/ 2 expressing the Joule-Lenz law [see Eq. (5.39)1. 

The Wiedemann-Franz Law. It is known from experiments that 
in addition to their high electrical conductivity, metals are distin- 
guished by a high thermal conductivity. The German physicists 


Classical Theory of Electrical Conductance of Metals 


233 


G. Wiedemann and R. Franz discovered an empirical law according to 
which the ratio of the thermal conductivity x to the electrical conduc- 
tivity a is about the same for all metals and changes in proportion 
to the absolute temperature. For example, for aluminium at 
room temperature, this ratio is 5.8 X 10“*, for copper it is 6.4 x 
X 10~ 8 , and for lead it is 7.0 X 10‘ 8 J*Q/(s-K). 

Non-metallic crystals are also capable of conducting heat. The 
thermal conductivity of metals, however, considerably exceeds that 
of dielectrics. It thus follows that the free electrons instead of the 
crystal lattice are responsible for the transfer of heat in metals. Con- 
sidering these electrons as a monatomic gas, we can adopt an expres- 
sion from the kinetic theory of gases for the thermal conductivity: 

1 

x'= y nmvlcy (11.1 1) 


(see Eq. (16.26) of Vol. I, p. 418; the density p has been replaced with 
the product nm, and ( v ) with i;]. The specific heat capacity of a mon- 

3 3 

atomic gas is c v = -j- ( RIM ) = -j- ( k/m ). Using this value in Eq. 
(11.11), we obtain 

x — y nkvl 


Dividing x by Eq. (11.9) for a and then substituting kT for 
ynw", we arrive at the expression 


, = W =3 (i) 2j , 


( 11 . 12 ) 


that expresses the Wiedemann-Franz law. 

Introduction oi the numerical values of k and e into Eq. (11.12) 
yields 

-£- = 2.23x10-* T 


When T = 300 K, we get the value 6.7 X 10~ 6 J-fi/(s-K) for x/a, 
which agrees quite well with experimental data (see the values of 
x/a given above for aluminium, copper, and lead). It was later estab- 
lished, however, that such a good coincidence is accidental, because 
when H. Lorentz performed the calculations more accurately, taking 
into account the distribution of the electrons by velocities, the value 
of 2 (Ar/e) 2 T was obtained for the ratio x/a, and it does not agree so 
well with the data of experiments. 

Thus, the classical theory was able to explain Ohm’s and the Joule- 
Lenz laws, and also gave a qualitative explanation of the Wiede- 
mann-Franz law. At the same time, this theory encountered quite 
appreciable difficulties. They include two basic ones. It can be seen 


234 


Electricity and Magnetism 


from Eq. (11.9) that the resistance of metals (i.e. the quantity that 
is the reciprocal of a) must increase as the square root of T. Indeed, 
we have no grounds to assume that the quantities n and l depend on 
the temperature. The velocity of thermal motion, on the other hand, 
is proportional to the square root of T . This theoretical conclusion 
contradicts experimental data according to which the electrical re- 
sistance of metals grows in proportion to the first power of T , i.e. 

j_ 

more rapidly than T 2 [see expression (5-24)1. 

The second difficulty of the classical theory is that an electron gas 

3 

must have a molar heat capacity equal to -y R- Adding this quantity 
to the heat capacity of the lattice, which is 3 R [see Eq. (13.1) of 
Vol. I, p. 375], we get the value oi ^ R for the molar heat capacity 

of a metal. Thus, in accordance with the classical electron theory, 
the molar heat capacity of metals ought to be 1.5 times higher than 
that of dielectrics. Actually, however, the heat capacity of metals 
does not differ appreciably from that of non-metallic crystals. Only 
the quantum theory of metals was able to explain this discrepancy. 


11.3. The Hall Effect 

If a metal plate through which a steady electric current is flowing 
is placed in a magnetic field perpendicular to it, then a potential 
difference of £/h = <Pi — <p 2 (FigT- 11.2) is set up between the plate 


/ 



Fig. 11.2 Fig. 11.3 


faces parallel to the directions of the current and field. This phenom- 
enon was discovered by the American physicist E. Hall in 1879 and 
is called the Hall effect or the galvanomagnetic effect. 

The Hall potential difference is determined by the expression 

U H = R n bjB (11.13) 





Classical Theory of Electrical Conductance of Metals 


235 


Here b = width of the plate 
/ =s current density 
B = magnetic induction of the field 
i?H = constant of proportionality known as the Hall coefficient. 

The Hall effect is easily explained by the electron theory. In the 
absence of a magnetic field, the current in the plate is due to the elec- 
tric field E 0 (Fig. 11.3). The equipotential surfaces of this field form 
a system of planes perpendicular to the vector E 0 . Two of them are 
shown in the figure by solid straight lines. The potential at all the 
points of each surface and, consequently, at points 1 and 2 too is the 
same. The current carriers — electrons — have a negative charge, there- 
fore the velocity of their ordered motion u is directed oppositely 
to the current density vector j. 

When the magnetic field is switched on, each carrier experiences 
the magnetic force F directed along side b of the plate and having 
a magnitude of 

F = euB (11.14) 

As a result, the electrons acquire a velocity component directed to- 
ward the upper (in the figure) face of the plate. A surplus of negative 
charges is formed at this face and, accordingly, a surplus of positive 
charges at the lower face. Consequently, an additional transverse 
electric field E b is produced. When the strength of this field reaches 
a value such that its action on the charges balances the force given 
by Eq. (11.14), a stationary distribution of the charges in a transverse 
direction will set in. The corresponding value of E B is determined 
by the condition eE B = euB . Hence, 

E B = uB 

The field E B adds to the field Eq to form the resultant field E. 
The equipotential surfaces are perpendicular to the field strength 
vector. Consequently, they will turn and occupy the position shown 
by the dash line in Fig. 11.3. Points 1 and 2 which were formerly on 
the same equipotential surface now have different potentials. To 
find the voltage appearing between these points, the distance b be- 
tween them must be multiplied by the strength E B : 

== bE B == buB 

Let us express u through /, n , and e in accordance with the equation 
/ = neu. The result is 

U a = — bjB 

a ne ’ 

Equations (11.15) and (11.13) coincide if we assume that 

1 


(11.15) 


236 


Electricity and Magnetism 


Inspection of Eq. (11.16) shows that by measuring the Hall coeffi- 
cient, we can find the concentration of the current carriers in a given 
metal (i.e. the number of carriers per unit volume). 

An important characteristic of a substance is the mobility of the 
current carriers in it. By the mobility of the current carriers is meant 



Fig. 11.4 

the average velocity acquired by the carriers at unit electric field 
strength. If the carriers acquire the velocity u in a field of strength 
E , then their mobility u 0 is 

Wo = -|- (11.17) 

The mobility can be related to the conductivity a and to the carrier 
concentration n . For this purpose, let us divide the equation ; = 
= neu by the field strength E. Taking into account that jlE = a 
and ulE = u 0 , we get 

a = neu 0 (1 1.18) 

Having measured the Hall coefficient i? H and the conductivity 
a, we can use Eqs. (11.16) and (11.18) to find the concentration and 
mobility of the current carriers in the relevant specimen. 

The Hall effect is observed not only in metals, but also in semicon- 
ductors. The sign of the effect can be used to see whether a semiconduc- 
tor belongs to the n- or p-type*. Figure 11.4 compares the Hall 
effect for specimens with positive and negative carriers. The direction 
of the magnetic force is reversed both when the direction of motion 
of the charge changes and when its sign is reversed. Hence, when the 
current and field have the same direction, the magnetic force exerted 
on positive and negative carriers has the same direction. Therefore, 
with positive carriers, the potential of the upper (in the figure) face 
is higher than that of the lower one, and with negative carriers the 
potential is lower. We can thus establish the sign of the current 
carriers after determining that of the Hall potential difference. 

It is of interest to note that in some metals the sign of U H cor- 
responds to positive current carriers. This anomaly is explained by 
the quantum theory. 

* In n-type semiconductors, the current carriers are negative, and in p- 
tvpe ones they are positive (see Vol. III). 




CHAPTER 12 ELECTRIC CURRENT 

IN GASES 


12.1. Semi-Self-Sustained 

and Self-Sustained Conduction 

The passage of an electric current through gases is called a gas 
discharge. Gases in their normal state are insulators, and current 
carriers are absent in them. Only when special conditions are created 
in gases can current carriers appear in them (ions, electrons), and an 
electric discharge be produced. 

Current carriers may appear in gases as a result of external action 
not associated with the presence of an electric field. In this case, the 
gas is said to have semi-self-sustained conduction. Semi-self-sustained 
discharge may be due to heating of a gas (thermal ionization), the 
action of ultraviolet rays or X-rays, and also to the action of radiation 
of radioactive substances. 

If the current carriers appear as a result of processes due to an 
electric field being produced in a gas, the conduction is called self- 
sustained. 

The nature of a gas discharge depends on many factors: on the 
chemical nature of the gas and electrodes, on the temperature and 
pressure of the gas, on the shape, dimensions, and mutual arrange- 
ment of the electrodes, on the voltage applied to them, on the den- 
sity and power of the current, etc. This is why a gas discharge may 
have very diverse forms. Some kinds of discharge are attended by a 
glow and sound effects — hissing, rustling, or crackling. 


12.2. Semi-Self-Sustaincd Gas Discharge 

Assume that a gas between electrodes (Fig. 12.1) continuously 
experiences a constant in intensity action of an ionizing agent (for 
example, X-rays). The action of the ionizer results in one or more 
electrons being detached from some of the gas molecules. The latter 
thus become positively charged ions. At not very low pressures, the 
detached electrons are usually captured by neutral molecules, which 
thus become negatively charged ions. Let A stand for the number of 
pairs of ions appearing under the action of the ionizer in unit volume 
per second. 

The process of ionization in a gas is attended by recombination 
of the ions, i.e. neutralization of unlike ions when they meet or the 

fnrmfltinn of ft nontral mAlonulo Kvr « «/»«!♦««« **- J — 



238 


Electricity and Magnetism 


The probability of two ions of opposite signs meeting each other is 
proportional to the number of both positive and negative ions. Hence 
the number of pairs of ions A n r recombining in unit volume per 
second is proportional to the square of the number 
of pairs of ions n per unit volume: 

A n T = rn 2 (12.1) 

(r is a constant of proportionality). 

In a state of equilibrium, the number of appear- 
ing ions equals the number of recombining ones, 
hence. 

An, = rn 2 (12.2) 

We thus get the following expression for the equi- 
librium concentration of ions (the number of pairs 
of ions in unit volume): 

n =Y ( 12 - 3 ) 

Several pairs of ions appear a second in 1 cm 3 of atmospheric air 
under the action of cosmic radiation and traces of radioactive sub- 
stances in the Earth’s crust. The constant r for air is 1.6 X 10”* cm 3 /s. 
Introduction of these values into Eq. (12.3) gives a value of about 
10 3 cm” 3 for the equilibrium concentration of ions in the air. This 
concentration is not adequate for the conduction to be noticeable. 
Pure dry air is a very good insulator. 

If we feed a voltage to electrodes, the ions will decrease in number 
not only because of recombination, but afeo because of the ions being 
drawn off by the field to the electrodes. Assume that A/ij pairs of 
ions are drawn off from unit volume every second. If the charge of 
each ion is e\ then the neutralization of one pair of ions on the elec- 
trodes is attended by the transfer of the charge e ' along the circuit. 
Every second, An^Sl pairs of ions reach the electrodes (here S is 
the area of the electrodes, l is the distance between them; the pro- 
duct SI equals the volume of the space between the electrodes). Con- 
sequently, the current in the circuit is 

I = e A/ijiSf 

whence 

An J=-7i ( 12 - 4 > 


0«*- 


EL 

Fig. 12.1 


where j is the current density. 

When a current is present, the condition of equilibrium is as fol- 
lows: 


Arii = A/ir + Ani 


Electric Current in Gates 


239 


Substituting for An, and Anj their values from Eqs. (12.1) and (12.4), 
we arrive at the equation 

AR t = rn 2 -f -L- (12.5) 

The current density is determined by the expression 

/ = e'n(< + «;)£ (12.6) 

where and uj are the mobilities of the positive and negative ions, 
respectively [see Eq. (11.17)1. 

Let us consider two extreme cases — weak and strong fields. 
With weak fields, the current density will be very small, and the 
addend jle'l in Eq. (12.5) may be disregarded in comparison with 
rn 2 (this signifies that the ions leave the space between the electrodes 
mainly as a result of recombination). Equation (12.5) thus trans- 
forms into Eq. (12.2), and we get Eq. (12.3) for the equilibrium con- 
centration of the ions. Using this value of n in Eq. (12.6), we get 

/ = e' K + «;> E (12.7) 

The multiplier of E in Eq. (12.7) does not depend on the field strength. 
Hence, with weak fields, a semi-self-sustained gas discharge obeys 
Ohm’s law. 

The mobility of ions in gases has a value of the order of 
10 -4 (m-s _1 )/(V *m -1 ) tor 1 (cm • s -1 )/(V • cm -1 )]. Hence, at the equi- 
librium concentration n = 10* cm - * = 10“ m - *, and the field 
strength E = 1 V/m, the current density will be 

/= 1.6 X 10 -19 x 10»(10-*-h 10“*) X 1 ~ 10-“ A/m 2 = 10-“ A/cm 2 

(see Eq. (12.6); the ions are assumed to be singly charged]. 

With strong fields, we may disregard the addend rn 2 in Eq. (12.5) 
in comparison with jle'l. This signifies that virtually all the appear- 
ing ions will reach the electrodes without having time to recombine. 
In these conditions, Eq. (12.5) becomes 



whence 

/ = e'Aihl ( 12 . 8 ) 

This current density is produced by all the ions originated by the 
ionizer in a column of the gas with unit cross-sectional area between 
the electrodes. Consequently, this current density is the greatest at 
the given intensity of the ionizer and the given distance l between 
the electrodes. It is called the saturation current density /, at . 

Let us calculate / gat for the following conditions: Anj = 10 cm -3 = 
— 10 7 m~* (this is approximately the rate of ion formation in the 



240 


Electricity and Magnetism 


atmospheric air in ordinary conditions), / = 0.1 m. The introduction 
of these data into Eq. (12.8) yields 

U at = 1.6 X 10-» X 10 7 x 10-* - 10“ 13 A/m 2 = 10' 17 A/cm 2 

These calculations show that the conduction of air in ordinary con- 
ditions is negligibly small. 

At intermediate values of E , there is a smooth transition from a 
linear dependence of / on E to saturation; when the latter is reached, 
7 stops depending on E (see the solid curve in Fig. 12.2). The region 
of saturation is followed by a region of a sharp growth in the current 

(see the portion of the curve depicted by 
the dash line). The explanation of this 
growth is that beginning from a certain 
value of E , the electrons* given birth to 
by the external ionizer manage to acquire 
a considerable energy while on their free 
path. This energy is sufficient to ionize 
the molecules they collide with. The free 
electrons produced in this ionization, after 
gaining speed, cause ionization in their 
turn. Thus, an avalanche-like reproduc- 
tion of the primary ions produced by the 
external ionizer occurs, and the discharge 
current is amplified. The process does not lose its nature of a semi- 
self-sustained discharge, however, because after the action of the 
external ionizer stops, the discharge continues only until all the elec- 
trons (primary and secondary) reach the anode (the rear boundary 
of the space containing ionizing particles — electrons — moves to- 
ward the anode). For a discharge to become self-sustained, two meet- 
ing avalanches of ions are needed. This is possible only if ionization 
by a collision is capable of giving birth to carriers of both signs. 

It is very important that the semi-self-sustained discharge cur- 
rents amplified as a result of reproduction of the carriers are propor- 
tional to the number of primary ions produced by the external ioniz- 
er. This property of a discharge is used in proportional counters (see 
the following section). 



Fig. 12.2 


12.3. Ionization Chambers and Counters 

Ionization chambers and counters are employed for detecting and 
counting elementary particles, and also for measuring the intensity 
of X-rays and gamma rays. The functioning of these instruments is 
based on the use of a semi-self-sustained gas discharge. 

* Owing to the greater length of their free path, electrons acquire the ability 
to produce ionization bv a collision earlier than eras ions do. 



Electric Current in Gases 


241 


The schematic diagram of an ionization chamber and a counter 
is the same (Fig. 12.3). They differ only in their operating conditions 
and structural features. A counter (Fig. 12.36) consists of a cylindri- 
cal body along whose axis a thin wire (anode) fastened on insulators 


i 



Fig. 12.3 

is stretched. The body of the counter is the cathode. A window of 
mica or aluminium foil is made in the end of the counter to admit the 
ionizing particles. Some particles, and also X-rays and gamma rays 
penetrate into a counter or an ionization chamber directly through 



their walls. An ionization chamber (Fig. 12.3a) can have electrodes 
of various shapes. In particular, they may be the same as in a coun- 
ter, have the shape of plane parallel plates, etc. 

Assume that a high-speed charged particle producing N 0 pairs 
of primary ions (electrons and positive ions) flies into the space be- 
tween the electrodes. The ions produced are carried along by the field 
toward the electrodes, and as a result a certain charge q, which we 
shall call a current pulse, passes through resistor R . Figure 12.4 
shows how the current pulse q depends on the voltage U between the 
electrodes for two different amounts of primary ions N 0 differing by 



242 


Electricity and Magnetism 


three times {N 02 = 3N 01 ). Six regions can be earmarked on the graph. 
Regions I and II were considered in the preceding section. In parti- 
cular, region II is the region of the saturation current — all the ions 
produced by an ionizing particle reach the electrodes without having 
time to recombine. It is quite natural that the current pulse does not 
depend on the voltage in these conditions. 

Beginning from the value £/ p , the field strength becomes sufficient 
for the electrons to be able to ionize the molecules by a collision. 
Therefore, the number of electrons and positive ions grows like an 
avalanche. As a result, AN 0 ions reach each of the electrodes. The 
quantity A is called the gas amplification factor. In region III , this 
factor does not depend on the number of primary ions (but does de- 
pend on the voltage). Therefore, if we keep the voltage constant, the 
current pulse will be proportional to the number of primary ions. 
Region III is called the proportional region, and the voltage £/ p — 
the threshold of the proportional region. The gas amplification factor 
changes in this region from 1 at its beginning to 10M0 4 at its end 
(the scale along the g-axis has not been observed in Fig. 12.4; only 
the ratio of 1 : 3 between the ordinates in regions II and III has been 
observed). 

In region /F, called the region of partial proportionality, the gas 
amplification factor A depends to a greater and greater extent on 
N 0 , In this connection, the difference between the current pulses 
produced by different numbers of primary ions becomes smoothed 
out more and more. 

At voltages corresponding to region F (it is known as the Geiger 
region, and the voltage U g as the threshold of this region), the process 
acquires the nature of a self-sustained discharge. The primary ions 
only produce an impetus for its appearance. The current pulse in 
this region is absolutely independent of the number of primary ions. 

In region VI , the voltage is so high that a discharge, after once 
being set up, does not stop. It is therefore called the region of con- 
tinuous discharge. 

Ionization Chambers. An ionization chamber is an instrument op- 
erating without gas amplification, i.e. at voltages corresponding to 
region II, There are two kinds of ionization chambers. Chambers of 
one kind are used for registering the pulses initiated by individual 
particles (pulse chambers). A particle flying into the chamber pro- 
duces a certain number of ions in it, and as a result the current I 
begins to flow through resistor R. The result is that the potential of 
point 1 (see Fig. 12.3a) rises and becomes equal to IR (the initial 
potential of this point was the same as that of earthed point 2 ). 
This potential is fed to an amplifier, and after being amplified op- 
erates a counting device. After all the charges that have reached the 
inner electrode pass through resistor /?, the current stops and the 
potential of point 1 again becomes equal to zero. The nature of op- 



Electric Current in Gases 


243 


eration of the chamber depends on the duration of the current pulse 
set up by one ionizing particle. 

To determine what the duration of a pulse depends on, let us con- 
sider a circuit consisting of capacitor C and resistor R (Fig. 12.5). 
If we impart the opposite charges +<7o and — <7o the capacitor 
plates, a current will flow through resistor i?, and the charges 
on the plates will diminish. The instantaneous value of the voltage 
applied across the resistor is U — q/C . Hence, 
we get the following expression for the current: 

u a 


/ = 


RC 


(12.9) 




Let us substitute — dqldt for the current, where 
— dq is the decrement of the charge on the 
plates during the time dt . As a result, we get 
the differential equation 

dq q_ ^ dq __ i 

q 


J 


dt 


RC 


or 


RC 


dt 


Fig. 12.5 


According to Eq. (12.9), dqlq = dill . We can therefore write 

1 


dl 

I 


RC 


dt 


Integration of this equation yields 


In / — — 1 + In /<> 


(In I 0 is the integration constant). Finally^ raising the expression 
obtained to a power, we arrive at the equation 

/ = / 0 ex p(— (12.10) 

It is easy to see that 7 0 is the initial value of the current. 

It follows from Eq. (12.10) that during the time 

x = RC (12.11) 

the current diminishes to i/e of its original value. Accordingly, the 
quantity x is called the time constant of a circuit. The greater this 
quantity, the slower is the rate of diminishing of the current in a 
circuit. 

The diagram of an ionization chamber (see Fig. 12.3a)‘is similar to 
that shown in Fig. 12.5. The part of C is played by the interelectrode 
capacitance shown by a dash line on the diagram of the chamber. 
An increase in the resistance of R is attended by a growth in the volt- 
age across points 1 and 2 at a given current, and this, consequently, 
facilitates the registration of the pulses. This circumstance induces 



244 


Electricity and Magnetism 


designers to use the highest possible resistance of R . At the same 
time, for the chamber to be able to register separately the current 
pulses set up by particles rapidly following one another, the time con- 
stant must not be great. Therefore, designers have to make a compro- 
mise when choosing the resistance of R for pulse chambers. It is 
usually taken of the order of 10 s ohms. Hence, at C ~ 10“ u F, the 
time constant is 10~ a s. 

Another kind of ionization chamber is the so-called integrating 
chamber. The resistance of R in them is of the order of 10 13 ohms. 
At C ~ 10" 11 F, the time constant is 10 4 s. In this case, the current 
pulses produced by separate ionizing particles merge and a steady 
current flows through the resistor. Its magnitude characterizes the 
total charge of the ions produced in the chamber in unit time. Thus, 
the ionization chambers of these two kinds differ only in the value 
of the time constant RC . 

Proportional Counters. The pulses set up by separate particles can 
be amplified quite considerably (up to 10 3 -10 4 times) if the voltage 
between the electrodes is in region III (see Fig. 12.4). An instrument 
operating in such conditions is called a proportional counter. The 
anode of the counter is made in the form of a wire of several hund- 
redths of a millimetre in diameter. The field strength near the wire 
is especially high. With a sufficiently great voltage between the elec- 
trodes, the electrons produced near the wire acquire an energy under 
the action of the field that is adequate for producing ionization of the 
molecules by a collision. The result is reproduction of the ions. The 
dimensions of the space in which reproduction occurs increase with 
the voltage. The gas amplification factor grows accordingly. 

The number of primary ions depends on the nature and energy of 
the particles producing the pulse. Therefore, the magnitude of the 
pulses at the output of a proportional counter makes it possible to 
distinguish various particles, and also to sort particles of the same 
nature by their energies. 

Geiger-Miiller Counters. A still greater amplification of the pulse 
(up to 10 8 ) can be attained by making a counter function in the Geiger 
region (region V in Fig. 12.4). A counter operating in these conditions 
is called a Geiger-Miiller counter (or more briefly a Geiger counter). 
A discharge in the Geiger region, being “launched” by an ionizing 
particle, subsequently transforms into a self-sustained one. Hence, 
the magnitude of the pulse does not depend on the initial ionization. 
To obtain separate pulses from individual particles, the discharge 
produced must be rapidly interrupted (quenched). This is achieved 
either with the aid of an external resistance R (in non-self-quenching 
counters), or at the expense of processes appearing in the counter 
itself. In the latter case, the counter is called self-quenching. 

The quenching of a discharge with the aid of an external resistance 
is due to the fact that when a discharge current flows in the resistance, 



Electric Current in Gases 


245 


a great voltage drop is set up in it. Consequently, only part of the 
applied voltage falls to the lot of the interelectrode space, and it is 
insufficient for maintaining the discharge. 

Stopping of a discharge in self-quenching counters is due to the 
following reasons. Electrons have a mobility that is about 1000 times 
greater than the mobility of positive ions. Therefore, during the time 
it takes the electrons to reach the wire, the positive ions do not vir- 
tually move from their places. These ions produce a positive space 
charge that weakens the field near the wire, and the discharge stops. 
Quenching of the discharge in this case is prevented by additional 
processes which we shall not consider. To suppress them, an admix- 
ture of a polyatomic organic gas (for example, alcohol vapour) is 
added to the gas filling the counter (usually argon). Such a counter 
separates pulses from particles following one another with an inter- 
val of the order of 10” 4 s. 


12.4. Processes Leading to the Appearance 
of Current Carriers 
in a Self-Sustained Discharge 

Before commencing to describe the various kinds of self-sustained 
gas discharge, we shall consider the basic processes leading to the 
production of current carriers (electrons and ions) in such discharges. 

Collisions of Electrons with Molecules. The collisions of electrons 
(and also ions) with molecules can have an elastic or inelastic nature. 
The energy of a molecule (like that of an atom) is quantized. This 
signifies that it can have only discrete (i.e. separated by finite inter- 
vals) values called energy levels. The state with the smallest energy 
is called the ground one. To transfer a molecule from its ground state 
to various excited ones, definite values of the energy W x , W t , etc. 
are needed. A molecule can be ionized by imparting to it a suffi- 
ciently great energy W*. 

Upon transition to an excited state, a molecule usually stays in 
it only ~ 10" 8 s, after which it passes back to its ground state, 
emitting its surplus energy in the form of a quantum of light — a 
photon. Molecules can spend a considerably greater time (about 
10“ 8 s) in certain excited states called metastable. 

The laws of energy and momentum conservation must be obeyed 
when particles collide. Therefore, definite limitations are imposed on 
the transfer of energy in a collision — not all the energy which a col- 
liding particle has can be transferred to another particle. 

If in a collision, an energy sufficient for exciting a molecule cannot 
be imparted to it, the total kinetic energy of the particles remains 
unchanged, and the collision will be elastic. Let us find the energy 


246 


Electricity and Magnetism 


imparted to the particle that is struck in an elastic collision. Assume 
that a particle of mass m 1 having the velocity i; 10 collides with a 
stationary (v 20 = 0) particle of mass m 2 . The following conditions 
must be observed in a central collision: 

2 2 '2 


m iV l0 — m i v i + m 2 v 2 


where v x and v 2 are the velocities of the particles after the collision. 
The velocity of the second particle from these equations will be 


^2 


2 m l 

m i + m i 


ViO 


(see Sec. 3.H of Vol. I, p. 104 et seq.). 

The energy transmitted to the second particle in an elastic colli- 
sion is determined by the expression 

AW m 2 y a m i^io 4m 1 m 1 

el 2 2 


If <C m 2 , this equation is simplified as follows: 


AW el = 



4m 




10 


4 m 1 

m t 


( 12 . 12 ) 


where W 10 is the initial energy of the incident particle. 

It can be seen from Eq. (12.12) that a light particle (electron) in 
an elastic collision with a heavy particle (molecule) gives up to it 
only a small fraction of its stock of energy. The light particle “re- 
bounds” from the heavy one like a ball from a wall, and its velocity 
remains virtually unchanged in magnitude. The relevant calculations 
show that in a non-central collision the fraction of the energy trans- 
ferred is still smaller. 

With a sufficiently high energy of the incident particle (electron 
or ion), a molecule may be excited or ionized. In this case, the total 
kinetic energy of the particles is not conserved — part of the energy 
goes for excitation or ionization, i.e. for increasing the internal ener- 
gy of the colliding particles or for splitting one of the particles into 
two fragments. 

Collisions attended by the excitation of particles are called in- 
elastic collisions of the first kind. A molecule in an excited state upon 
colliding with another particle (electron, ion, or neutral molecule) 
can pass over to the ground state without emitting its surplus energy, 
but transferring it to this particle. As a result, the total kinetic ener- 
gy of the particles after the collision will be greater than before it. 
Such collisions are known as inelastic collisions of the second kind. 
Molecules pass over from a metastable state to the ground one as a 
result of collisions of the second kind. 


Electric Current in Gate* 


247 


In an inelastic collision of the first kind, the equations of energy 
and momentum conservation have the form 


'"i”! , , a 1x7 

2 ~~2 1 2 r 


(12.13) 


m t v i0 = m t v t -f m 2 v 2 


where is the increment of the internal energy of a molecule 

corresponding to its transition to an excited state. Deleting v 2 from 
these equations, we get 


AW^mt = nhv i 0 v 2 


m 1 2 


(12.14) 


At a given velocity of the striking particle (i> 10 ), the increment of 
the internal energy AW^t depends on the velocity v t with which the 
molecule travels after the collision. Let us find the greatest possible 
value of AW^int- To do this, we shall differentiate function (12.14) 
with respect to v 2 and equate the derivative to zero: 


du 2 


= m » v to 


m l 


m^u 2 


0 


Hence, v 2 = + m 2 ). Substitution of this value for v 2 in 

Eq. (12.14) yields 


Al^int. max 


”»» 

JTl\ -j- 2 


(12.45) 


If the incident particle is considerably>lighter than the struck one 
(m x < m 2 ), the factor mj + m 2 ) in Eq. (12.15) is close to unity. 
Thus, when a light particle (electron) strikes a heavy one (molecule), 
almost all the energy of the incident particle can be used to excite or 
ionize the molecule*. 

Even if the energy of the incident particle (electron) is sufficiently 
great, however, a collision does not necessarily result in the excita- 
tion or ionization of a molecule. These processes have definite proba- 
bilities depending on the energy (and, therefore, on the velocity) 
of the electron. Figure 12.6 shows the approximate path followed by 
these probabilities. The higher the velocity of the electron, the smaller 
is the duration of its interaction with the molecule near which it 
flies. Hence, both probabilities rapidly reach a maximum, and then 
diminish with an increase in the energy of the electron. Inspection 
of the figure shows that an electron having, for example, the energy 
W' will cause ionization of a molecule with greater probability than 
its excitation. 


* When ionization occurs, Eqs. (12.13) become more complicated because 
there will be three particles instead of two after a collision. Tne conclusion on 
the possibility of spending almost all of the electron's energy for ionization is 
correct, however. 


248 


Electricity and Magnetism 


Photoionization. Electromagnetic radiation consists of elementary 
particles called photons. The energy of a photon is fto>, where h is 
Planck’s constant divided by 2 ji [see Eq. (7.43)1, and <o is the cyclic 
frequency of the radiation. A photon can be absorbed by a molecule, 
and its energy goes to excite or ionize the molecule. In this case, the 
ionization of the molecule is called photoionization. Ultraviolet 
radiation is capable of producing direct photoionization. The energy 
of a photon of visible light is insufficient to detach an electron from 
a molecule. Hence, visible radiation is not capable of producing 

direct photoionization. It may be the 
cause, however, of so-called stepped 
photoionization. This process is car- 
ried out in two steps. In the first one, 
a photon transfers the molecule to 
an excited state. In the second step, 
the excited molecule is ionized as a 
result of its colliding with another 
molecule. 

Short-wave radiation may appear 
in a gas discharge that is capable of 
producing direct photoionization. 
A sufficiently fast electron may not 
only ionize a molecule when it col- 
lides with it, but also transfer the ion formed into an excited state. 
The transition of an ion to the ground state is attended by the emis- 
sion of radiation, having a higher frequency than that of a neutral 
molecule. The energy of a photon of such radiation is sufficient for 
direct photoionization. 

Emission of Electrons by the Surface of Electrodes. Electrons may 
be supplied to a gas-discharge space as a result of their emission by 
the surface of the electrodes. Such kinds of emission as thermionic 
(thermoelectron), secondary electron, and autoelectronic emission 
play the main part in some kinds of discharge. 

Thermionic emission is the name given to the emission of electrons 
by heated solid or liquid bodies. Owing to the free electrons in a 
metal having a variety of velocities in accordance with a distribution 
law, there is always a certain number of them whose energy is suffi- 
cient for them to overcome the potential barrier and leave the metal. 
The number of such electrons at room temperature is negligibly small. 
With elevation of the temperature, however, the number of electrons 
capable of leaving the metal grows very rapidly and becomes quite 
noticeable at a temperature of the order of 1000 K. 

By secondary electron emission is meant the emission of electrons 
by the surface of a solid or a liquid body when it is bombarded with 
electrons or ions. The ratio of the number of emitted {secondary) 
electrons to the number of particles producing the emission is called 



Energy of electron 

Fig. 12.6 


Electric Current in Gases 


249 


the secondary electron emission coefficient. When electrons are used 
to bombard the surface of a metal, the values of this coefficient vary 
from 0.5 (for beryllium) to 1.8 (for platinum). 

Autoelectronic (or cold) emission is the emission of electrons by the 
surface of a metal occurring when an electric field of a very high 
strength (~ 10 8 V/m) is set up near the surface. This phenomenon is 
also sometimes called field-induced electron emission. 


12.5. Gas-Discharge Plasma 

Some kinds of self-sustained discharge are characterized by a very 
high degree of ionization. A highly ionized gas, provided that the 
total charge of the electrons and ions in each elementary volume 
equals (or almost equals) zero, is called a plasma. A plasma is a spe- 
cial state of a substance. The matter in the interior of the Sun and 
other stars having a temperature of scores of millions of kelvins is 
in this state. A plasma produced owing to the high temperature of 
a substance is called high-temperature (or Isothermal). A gas-discharge 
plasma, as its name implies, is one produced in a gas discharge. 

For a plasma to be in a stationary state, processes are needed that 
replenish the stock of ions diminishing as a result of recombination. 
In high-temperature plasma, this is achieved as a result of thermal 
ionization, in gas-discharge plasma, as a result of collision ioniza- 
tion by electrons accelerated by an electric field. The ionosphere 
(one of the layers of the atmosphere) is a special variety of plasma. 
The high degree of ionization of the molecules (~ 1%) is main- 
tained in the ionosphere by photoionization due to the Sun’s short- 
wave radiation. 

The electrons in a gas-discharge plasma participate in two mo- 
tions — chaotic with a certain average velocity (v) and ordered motion 
in a direction opposite to E with the average velocity (u) much 
smaller than (v). 

We shall prove that an electric field not only leads to ordered 
motion of the electrons of a plasma, but also increases the velocity 
(n> of their chaotic motion. Assume that at the moment when the 
field is switched on the gas contains a certain number of electrons 
whose average velocity corresponds to the gas temperature 

= ftTg). In the interval between two successive 

collisions with molecules, an electron covers on an average the path 
l (Fig. 12.7; the trajectory of the electron is curved slightly under 
the action of the force — eE). The work done by the field on the elec- 
tron is 

A = eEl P 


(12.16) 


250 


Electricity and Magnetism 


where l F is the projection of the electron’s path onto the direction of 
the force exerted on it. Owing to collisions with molecules, the di- 
rection of motion of the electron constantly changes chaotically. 
The magnitude and sign of l F change accordingly. This is why the 
work given by Eq. (12.16) for separate portions of the path varies in 
magnitude and changes in its sign. On some sections, the field in- 
creases the energy of the electron, on others diminishes it. If ordered 
motion of the electrons were absent, the average value of l F and, 
consequently, the work given by Eq. (12.16) would be zero. The 

presence of ordered motion, however, leads 
to the average value of the work A differ- 
ing from zero; it is positive and equals 

<A)-eE{u)x = eE(u)Jfr (12.17) 

where x is the average time needed by 
the electrons to cover their free path 
( ( u > < (i>>). 

Thus, a field on an average increases the 
energy of the electrons. True, an electron 
upon colliding with a molecule gives up 
part of its energy to it. But, as we have seen in the preceding section, 
the fraction 6 of the energy transferred in an elastic collision is very 
small — it averages* (6> = 2 {ml M) (here m is the mass of an elec- 
tron, and M that of a molecule). 

In a rarefied gas (in which l is greater) and with a sufficiently great 
field strength E y the work (A) [Eq. (12.17)] may exceed the energy 

-~m(i^)-<6> transferred on an average to a molecule in each col- 
lision. The result will be a growth in the energy of chaotic motion of 
the electrons. It ultimately reaches values sufficient to excite or 
ionize a molecule. Beginning from this moment, part of the colli- 
sions stop being elastic and are attended by a large loss of energy. 
Therefore, the average fraction (6) of energy transferred increases. 

Thus, the electrons acquire the energy needed for ionization not 
during one interval between collisions, but gradually in the course 
of a number of them. Ionization leads to the appearance of a large 
number of electrons and positive ions — a plasma is produced. 

The energy of the electrons of a plasma is determined by the con- 
dition that the average value of the work done by the field on an 
electron during one interval between collisions equals the average 
value of the energy given up by the electron upon colliding with a 



* According to Eq. (12.12), in a central collision 6 = 4 (mlM). When the 
electron and the molecule only slightly touch each other, we have 6 « 0. 



Electric Current in Gases 


251 


molecule: 

eEiuy^^P-m 

Here (6> is an intricate function of (v). 

Experiments show that the Maxwell distribution by velocities 
holds for the electrons in a gas-discharge plasma. Owing to the weak 
interaction of the electrons with the molecules (in an elastic collision 
6 is very small, while the relative number of inelastic collisions is 
negligible), the average velocity of chaotic motion of the electrons 
is many times greater than the velocity corresponding to the tempe- 
rature To of the gas. If we introduce the temperature of the electrons 

T t determining it from the equation -m(v 2 ) = ~ 2 ~kT e , then we 

get a value of the order of several tens of thousands of kelvins for 
T e . The failure of the temperatures T g and T e to coincide indicates 
that there is no thermodynamic equilibrium between the electrons 
and molecules in a gas-discharge plasma*. 

The concentration of the current carriers in a plasma is very high. 
Therefore, a plasma is an excellent conductor. The mobility of the 
electrons is about three orders of magnitude greater than that of the 
ions. Hence, the current in a plasma is mainly set up by its electrons. 


12.6. Glow Discharge 

A glow discharge appears at low pressures. It can be observed in 
a glass tube about 0.5 m long with flat metal electrodes soldered into 
its ends (Fig. 12.8). A voltage of ~ 1000 V is supplied to the 
electrodes. There is virtually no current in the tube at atmospheric 
pressure. If the pressure is lowered, then approximately at 50 mmHg 
a discharge appears in the form of a glowing sinuous thin cord con- 
necting the anode and the cathode. Lowering of the pressure is attend- 
ed by thickening of the cord, and at about 5 mmHg the cord fills 
the entire cross section of the tube— a glow discharge sets in. Its 
principal parts are shown in Fig. 12.8. Near the cathode is a thin 
luminous layer called the cathode luminous film. Between the cath- 
ode and the luminous film is the Aston dark space. At the other side 
of the luminous film is a weakly luminous layer which by contrast 
appears to be dark and is accordingly known as the cathode (or 
Crookes) dark space. This layer bounds on a luminous region called 
the negative glow. All the above layers form the cathode part of the 
glow discharge. 


* The average energy of the molecules, electrons, and ions in a high-temper- 
ature plasma is the same* This explains its other »name- isothermal olasma. 



252 


Electricity and Magnetism 


The negative glow is followed by the Faraday dark space. The 
boundary between them is blurred. The remaining part of the tube 
is filled with a luminous gas; it is called the positive column. At 
a lower pressure, the cathode part of the discharge and the Faraday 
dark space become wider, while the positive column becomes shorter. 
At a pressure of the order of 1 mmHg, the positive column breaks 
up into a number of alternating dark and light bent layers— strata. 

Measurements made with the aid of probes (thin wires soldered in 
at different points along the tube) and by other means have shown that 
the potential changes non-uniformly along a tube (see the graph in 


Aston Cathode Faradau 


Cathode 




Cathode film 

f 


Cathode potential drop 

Fig. 12.8 


Dark spaces 



Anode 


Negate ve glow Positive column ^region^ S 



Fig. 12.8). Virtually the entire potential drop falls to the share of the 
first three parts of the discharge up to the cathode dark space inclu- 
sively. This portion of the voltage applied to a tube is called the 
cathode potential drop. The potential remains unchanged in the region 
of the negative glow — here the field strength is zero. Finally, the 
potential gradually grows in the Faraday dark space and in the po- 
sitive column. Such a distribution of the potential is due to the for- 
mation in the cathode dark space of a positive space charge be- 
cause of the increased concentration of the positive ions. 

The main processes needed to maintain a glow r discharge occur in 
its cathode part. The other parts of the discharge are not significant, 
they may even be absent (with a small spacing of the electrodes or 
at a low pressure). There are two main processes — secondary elec- 
tron emission from the cathode produced by its bombardment with 
positive ions, and collision ionization of the gas molecules by elec- 
trons. 

The positive ions accelerated by the cathode potential drop bom- 
bard the cathode and knock electrons out of it. These electrons are 
Kv tho olpftt.Hr. field in the Aston dark space. Acauiring 



Electric Current in Gate* 


253 


sufficient energy, they begin to excite the gas molecules, owing to 
which the cathode luminous film appears. The electrons that fly 
without any collisions into the region of the cathode dark space have 
a high energy, and as a result they ionize the molecules more fre- 
quently than they excite them (see the graphs in Fig. 12.6). Thus, 
the intensity of glowing of the gas diminishes, but in return many 
electrons and positive ions appear. The ions produced first have a 
very low velocity. As a result, a positive space charge is formed in 
the cathode dark space. This leads to redistribution of the potential 
along the tube and to the appearance of the cathode potential drop. 

The electrons appearing in the cathode dark space penetrate into 
the negative glow region that is characterized by a high concentra- 
tion of electrons and positive ions and by a total space charge close 
to zero (a plasma). Therefore, the field strength here is very low. Ow- 
ing to the high concentration of electrons and ions, an intensive re- 
combination process goes on in the negative glow region. It is attend- 
ed by the emission of the energy liberated during this process. Thus, 
the negative glow is mainly a glow of recombination. 

The electrons and ions penetrate from the negative glow region 
into the Faraday dark space because of diffusion (there is no field on 
the boundary between these regions, but in return there is a high 
gradient of electron and ion concentration). The lower concentration 
of the charged particles greatly diminishes the probability of recom- 
bination in the Faraday dark space. This is why the latter space seems 
to be dark. 

A field is already present in the Faraday dark space. The electrons 
carried away by this field gradually accumulate energy .so that the 
conditions needed for the existence of a plasma finally appear. The 
positive column is a gas-discharge plasma. It plays the part of a con- 
ductor joining the anode to the cathode parts of the discharge. The 
glow of the positive column is mainly due to transitions of excited 
molecules to their ground state. Molecules of different gases emit ra- 
diation of different wavelengths in such transitions. Therefore, the 
glow of the positive column has a characteristic colour for each gas. 
This circumstance is taken advantage of in glow tubes for manufac- 
turing luminous inscriptions and advertisements. These inscriptions 
are the positive column of a glow discharge. Neon gas-discharge 
tubes produce a red glow, argon ones a bluish-green glow, etc. 

If the electrode spacing is gradually diminished, the cathode part 
of the discharge remains unchanged whereas the length of the positive 
column diminishes until this column disappears completely. Next, 
the Faraday dark space disappears, and the length of the negative 
glow begins to decrease, the position of the boundary of this glow 
with the cathode dark space remaining unchanged. When the dis- 
tance from the anode to this boundary becomes very small, the 
discharge stops. 


254 


Electricity and Magnetism 


If the pressure is gradually lowered, the cathode part of the dis- 
charge extends over a greater and greater part of the interelectrode 
space, and finally the cathode dark space extends over almost the 
entire tube. The glow of the gas in this case stops being noticeable, 
but in return the tube walls begin to glow with a greenish colour! 
The majority of the electrons knocked out of the cathode and accel- 
erated by the cathode potential drop reach the tube walls without col- 
liding with molecules of the gas and cause the walls to glow upon 
striking them. For historical reasons, the stream of electrons emitted 
by the cathode of a gas-discharge tube at very low pressures was called 
cathode rays. The glow produced by bombardment with fast elec- 
trons is called cathodoluminescence. 

If a narrow canal is made in the cathode of a gas-discharge tube, 
part of the positive ions penetrate into the space beyond the cathode 
and form a sharply bounded beam of ions called canal (or positive) 
rays. Beams of positive ions were first obtained in exactly this way. 


12.7. Arc Discharge 

In 1802, the Russian physicist Vasili Petrov (1761-1834) discov- 
ered that when contacting carbon electrodes connected to a large gal- 
vanic battery are moved apart, a concentrated light flares up be- 
tween the electrodes. When the electrodes are horizontal, the heated 
luminescent gas bends in the shape of an arc. This is why the pheno- 
menon discovered by Petrov was called an electric arc. The current 
in the arc may reach enormous values (from 10 3 to 10 4 A) at a voltage 
of several scores of volts. 

An arc discharge can proceed at both a low (of the order of several 
millimetres of mercury) and a high (up to 1000 atmospheres) pressure. 
The main processes maintaining the discharge are thermionic emis- 
sion from the heated cathode surface and thermal ionization of the 
molecules due to the high temperature of the gas in the space be- 
tween the electrodes. Almost the entire interelectrode space is filled 
with a high-temperature plasma. It is the conductor through which 
the electrons emitted by the cathode reach the anode. The tempera- 
ture of the plasma is about 6000 K. In a superhigh-pressure arc, the 
temperature of the plasma may reach 10 000 K (we remind our reader 
that the temperature of the Sun’s surface is 5800 K). Owing to bom- 
bardment by positive ions, the cathode is heated to about 3500 K. 
The anode, bombarded by a powerful stream of electrons, is heated 
still more. As a result, the anode intensively evaporates, and a de- 
pression — a crater — is formed on its surface. The crater is the bright- 
est place in an arc. 

An arc discharge has a dropping volt-ampere characteristic 
(Fig. 12.9). The explanation is that a current increase is attended by 


Electric Current in Gates 


255 


a growth in the thermionic emission from the cathode and in the 
degree of ionization of the gas-discharge space. As a result, the resis- 
tance of this space diminishes at a greater rate than that of the cur- 
rent increase. 

Apart from the thermionic arc described above (i.e. a discharge due 
to thermionic emission from the heated surface of the cathode) an 
arc with a cold cathode is also encountered. 

Usually liquid mercury poured into a cylinder 
from which the air has been evacuated is the 
cathode of such an arc. The discharge occurs in 
the mercury vapour. The electrons fly out of the 
cathode as a result of autoelectronic emission. 

The strong field at the cathode surface needed 
for this to occur is set up by the positive space 
charge formed by the ions. The electrons are 
emitted not by the entire surface of the cathode, 
but by a small luminous and continuously moving cathode spot. The 
temperature of the gas in this case is not high. The molecules in the 
plasma are ionized, as in a glow discharge, as a result of collisions 
with the electrons. 



12.8. Spark and Corona Discharges 

A spark discharge is produced when the electric field strength reach- 
es the breakdown value E hr for the given gas. The value of E hr de- 
pends on the gas pressure; it is about 3 MV/m (30 kV/cm) for air. 
The value of E hT varies with the pressure. According to the experi- 
mentally established Paschen law, the ratio of the breakdown field 
strength to the pressure is approximately constant: 

« const 
P 

A spark discharge is attended by the formation of a brightly lu- 
minous tortuous branched canal along which a short-time strong- 
current pulse flows. An example is lightning; its length may be up to 
10 km, the diameter of the canal up to 40 cm, the current may reach 
100 000 and more amperes, and the duration of the pulse is about 
10" 4 s. Every stroke of lightning consists of several (up to 50) pulses 
flowing along the same canal; their total duration (together with 
the intervals between the pulses) may reach several seconds. The 
temperature of the gas in the spark canal is up to 10 000 K. The 
rapid strong heating of the gas leads to a sharp growth in the pressure 
and the production of shock and sound waves. This is why a spark 
discharge is attended by sound phenomena— from a weak crackling 
for a low-power spark to peals of thunder accompanying a stroke of 
lightning. 



256 


Electricity and Magnetism 


The appearance of a spark is preceded by the formation in the gas 
of a greatly ionized canal known as a streamer. The latter is obtained 
by overlapping of the separate electron avalanches appearing along 
the path of the spark. The forefather of each avalanche is an electron 
released by photoionization. How a streamer develops is shown in 
Fig. 12.10. Assume that the field strength has a value such that an 
electron flying out of the cathode as a result of some process or other 
acquires an energy sufficient for ionization along its free path. This 
causes multiplication of the electrons to occur— an avalanche is 
formed (the positive ions appearing during this process do not play a 
noticeable part owing to their much smaller mobility; they only set up 




the space charge resulting in redistribution of the potential). The 
short-wave radiation emitted by an atom that lost one of its inner 
electrons when ionized (this radiation is shown by wavy lines in the 
figure) produces photoionization of the molecules, the detached elec- 
trons giving birth to more and more new avalanches. After overlapping 
of the avalanches, a well-conducting canal — a streamer — is formed 
along which a powerful stream of electrons flows from the cathode to 
the anode — breakdown occurs. 

If the electrodes have a shape at which the field in the space be- 
tween them is approximately homogeneous (for example, they are 
spheres of a sufficiently great diameter), then breakdown occurs at 
a quite definite voltage t/ br whose value depends on the distance be- 
tween the spheres l ( U br = E bT l). This underlies the design of a 
spark voltmeter used to measure high voltages (from 10 3 to 10 5 V). 
During such measurements, the maximum distance l max I s determined 
at which a spark appears. Next multiplying Z? br by / max , we get 
the value of the voltage being measured. 

If one of the electrodes (or both) has a very great curvature (for 
example, the electrode is a thin wire or a sharp point), then when the 
voltage is not too high, a so-called corona discharge is produced. When 
the voltage grows, this discharge transforms into a spark or an arc 
discharge. 

In a corona discharge, the ionization and excitation of the molecu- 
les occur not in the entire interelectrode space, but only near an elec- 
trode having a small radius of curvature, where the field strength 
reaches values equal to or greater than E br . The gas glows in this 
part of the discharge. The glow has the form of a corona surrounding 



Electric Current in Gases 


257 


the electrode, and this explains the name given to this kind of dis- 
charge. A corona discharge from a point has the form of a luminous 
brush, and for this reason it is sometimes known as a brush discharge. 
Positive and negative coronas are distinguished depending on the 
sign of the corona electrode. The external corona region is between 
the corona layer and the non-corona electrode. Breakdown conditions 
(E E hr ) exist only within the limits of the corona layer. We can 
therefore say that a corona discharge is incomplete breakdown of the 
gas space. 

With a negative corona, the phenomena at the cathode are similar 
to those at the cathode of a glow discharge. The positive ions accel- 
erated by the field knock electrons out of the cathode. These elect- 
rons produce ionization and excitation of the molecules in the corona 
layer. In the external region of the corona, the field is not sufficient 
to impart the energy needed for ionization or excitation of the mole- 
cules to the electrons. For this reason, the electrons that penetrate 
into this region drift toward the anode under the action of the field. 
Part of the electrons are captured by the molecules, the result being 
the formation of negative ions. Thus, the current in the external 
region is due only to negative carriers — electrons and negative ions. 
The discharge in this region is of a semi-self-sustained nature. 

In a positive corona, the electron avalanches are conceived at the 
outer boundary of the corona and fly toward the corona electrode — 
the anode. The appearance of electrons giving birth to avalanches is 
due to photoionization produced by the radiation of the corona layer. 
The current carriers in the external region of the corona are the 
positive ions that drift to the cathode under the action of the field. 

If both electrodes have a great curvature (two corona electrodes), 
processes occur near each of them that are characteristic of a corona 
electrode of the given sign. Both corona layers are separated by an 
external region in which opposite streams of positive and negative 
current carriers travel. Such a corona is called a bipolar one. 

The self-sustained gas discharge mentioned in Sec. 12.3 when treat- 
ing counters is a corona discharge. 

The thickness of the corona layer and the discharge current grow 
with an increasing voltage. At a low voltage, the size of the corona 
is small, and its glow is hard to notice. Such a microscopic corona is 
produced near a sharp point off which an electric wind flows (see 
Sec. 3.1). 

The bluish electrical glow caused by corona discharge on masts and 
other high parts of a ship at sea before and after electrical storms was 
called St. Elmo’s fire in olden days. 

In high-voltage facilities, for example, in high-tension transmis- 
sion lines, a corona discharge leads to the harmful leakage of current. 
Measures therefore have to be taken to prevent it. For this purpose, 
for instance, the wires of high-tension lines are taken of a sufficiently 



258 


Electricity and Magnetism 


large diameter, which is the greater, the higher is the voltage of the 
line. 

The corona discharge has found a useful application in engineer- 
ing in electrical filters. The gas being purified flows through a tube 
along whose axis a negative corona electrode is arranged. The nega- 
tive ions present in a great number in the external region of the co- 
rona settle on the particles or droplets polluting the ga3 and are car- 
ried along with them to the external non-corona electrode. Upon 
reaching the latter, the particles become neutralized and settle on 
it. Later, blows are struck at the tube and the sediment formed by 
the precipitated particles drops into a collector. 


CHAPTER 13 ELECTRICAL 

OSCILLATIONS 


13.1. Quasistationary Currents 

When considering electrical oscillations, we have to do with time- 
varying currents. Ohm’s law and Kirchhoffs rules following from it 
were established for a steady current. They also hold, however, for 
the instantaneous values of a varying current and voltage if the 
changes are not too fast. Electromagnetic disturbances propagate 
along a circuit with a tremendous speed equal to the speed of light c . 
Assume that the length of a circuit is l . If during the time x = Ifc 
needed for the transmission of a disturbance to the farthest point 
of a circuit, the current changes insignificantly, then the instantaneous 
values of the current in all the cross sections of the circuit will be 
virtually identical. Currents obeying this condition are called qua- 
slstationary. For periodically varying currents, the condition for a 
quasistationary state is 

*-T< T 

where T is the period of the changes. 

The delay for a circuit 3 m long is x = 10“® s. Thus, up to values 
of T of the order of 10" 6 s (which corresponds to a frequency of 10 6 Hz), 
the currents in Such a circuit may be considered quasistationary. A 
current of industrial frequency (v = 50 or 60 Hz) is quasistationary 
for circuits up to about 100 km long. 

The instantaneous values of quasistationajy currents obey Ohm’s 
law. Hence, Kirchhoffs rules also hold for them. 

In the following when studying electrical oscillations, we shall 
always assume that the currents we are dealing with are quasista- 
tionary. 


13.2. Free Oscillations in a Circuit 
Without a Resistance 

Electrical oscillations may appear in a circuit containing an in- 
ductance and a capacitance. Such a circuit is therefore called an 
oscillatory circuit. Figure 13.1a shows the consecutive stages of an 
oscillatory process in an idealized circuit containing no resistance. 



260 


Electricity and Magnetism 


Oscillations can be set up in the circuit either by supplying a cer- 
tain initial charge to the capacitor plates or by producing a current 
in the inductance (for example, by switching off the external magnet- 
ic field passing through the coil turns). Let us use the first method. 
We shall connect the capacitor to a source of voltage after discon- 
necting it from the inductance. The result will be the appearance of 
unlike charges +q and — q on the plates (stage 1). An electric field 

will be set up between the plates, and its energy will be ^ (q 2 /C) 

{see Eq. (4.5)1. If we next switch off the voltage source and connect 
the capacitor to the inductance, it will begin to discharge, and a 

Stages : 12345 
I I 



W=-Jr kx 2 -Jmx 2 y /77X 1 -Jr kx 2 

(b) 

Fig. 13.1 


current will flow through the circuit. The energy of the electric 
field will diminish as a result, but in return a constantly growing 
energy of the magnetic field set up by the current flowing through the 

inductance will appear. This energy is y LI 2 [see Eq. (8.37)1. 

Since the resistance of the circuit is zero, the total energy consist- 
ing of the energies of the electric and magnetic fields is not used for 
heating the wires and will remain constant*. Therefore at the moment 
when the voltage across the capacitor and, consequently, the energy 
of the electric field vanish, the energy of the magnetic field and, con- 
sequently, the current reach their maximum value (stage 2\ begin- 
ning from this moment, the current flows at the expense of the self- 
induced e.m.f.). After this, the current diminishes, and, when the 

* Strictly speaking, in such an idealized circuit, energy would be lost on the 
radiation of electromagnetic waves. This loss (pows with an increasing frequency 
of oscillations and when the circuit is more "‘open 1 ’. 




Electrical Oscillations 


261 


charges on the plates reach their initial value q , the current will va- 
nish (stage 5). Next, the same processes occur in the opposite direc- 
tion (stages 4 and 5). After them, the system returns to its initial 
state (stage 5), and the entire cycle repeats again and again. The 
charge on the plates, the voltage across the capacitor, and the current 
flowing in the inductance periodically change (i.e. oscillate) during 
the process. The oscillations are attended by mutual transformations 
of the electric and magnetic field energies. 

Figure 13.16 compares the oscillations of a 
spring pendulum with those in the circuit. The 
supply of charges to the capacitor plates cor- 
responds to bringing the pendulum out of its 
equilibrium position by exerting an external 
force on it and imparting the initial devia- 
tion x to it. The potential energy of elastic de- 

formation of the spring equal to y kx 2 is pro- 
duced. Stage 2 corresponds to passing of the 
pendulum through its equilibrium position. 

At this moment, the quasi-elastic force vanishes, and the pendulum 
continues its motion by inertia. By this time, the energy of the pendu- 
lum completely transforms into kinetic energy and is determined by 

the expression y mx*. We shall let our reader compare the further 
stages. 

It can be seen from a comparison of electrical and mechanical os- 
cillations that the energy of an electric field y ( q 2 lC ) is similar to 
the potential energy of elastic deformation, and the energy of a mag- 
netic field y LI 2 is similar to the kinetic energy. The inductance L 

plays the part of the mass m, and the reciprocal of the capacitance 

(1/C) the part of the spring constant k . Finally, the displacement x 

of the pendulum from its equilibrium position corresponds to the 

• • 

charge q , and the speed x to the current / = q. We shall see below 
that the analogy between electrical and mechanical oscillations also 
extends to the mathematical equations describing them. 

Let us find an equation for the oscillations in a circuit without a 
resistance (an L-C circuit). We shall consider the current charging 
the capacitor to be positive* (Fig. 13.2). Hence, by Eq. (5.1), 



* With such a choice of the direction of the current, the analogy between 

electrical and mechanical oscillations is more complete: q corresponds to the 
• • • 
speed x (with a different choice, — q corresponds to the speed x). 


♦ 


— JffiUb — 

L 

Fig. 13.2 


262 


Electricity and Magnetism 


Equation (5.27) of Ohm’s law for circuit 1-3-2 is 

IR — <p 4 <p 2 -|- % i% 


In our case, R = 0, <Pi — <p 2 — — q!C , and g 12 = g s — 
= — L (dl/dt). Introducing these values into Eq. (5.27), we get 



(13.1) 

* • 

Finally, replacing dl!dt with q [see Eq. (5.1)1, we get 



(13.2) 

If we introduce the symbol 


1 

< ° o= y~lc 

(13.3) 

Eq. (13.2) becomes 



(13.4) 


which is our good acquaintance from the science of mechanical oscil- 
lations [see Eq. (7.7) of Vol. I, p. 188]. The following function is 
a solution of this equation: 

q = q m cos (c o 0 t + a) (13.5) 

(the subscript “m” stands for maximum). 

Thus, the charge on the capacitor plates changes according to 
a harmonic law with a frequency determined by Eq. (13.3). This 
frequency is called the natural frequency of the circuit (it corresponds 
to the natural frequency of a harmonic oscillator). We get the so- 
called Thomson formula for the period of the oscillations: 

T = 2n V~LC (13.6) 

The voltage across the capacitor differs from the charge by the 
factor i/C: 

V = -^r* cos (co 0 * + a) = U m cos (< o 0 t + a) (13.7) 

Time differentiation of Eq. (13.5) yields an expression for the cur- 
rent: 

/ = — <D 0 ? m sin(w 0 < + a)==/ m cos (o» 0 f + a-fy) (13.8) 

Thus, the current leads the voltage across the capacitor in phase by 
ji/2. 

A comparison of Eqs. (13.5) and (13.7) with Eq. (13.8) shows that 
at the moment when the current reaches its maximum value, the 
charge and the voltage vanish, and vice versa. We have already 



Electrical Oscillations 


263 


established this relation between the charge and the current on the 
basis of energy considerations. 

Examination of Eqs. (i3.7) and (13.8) shows that 

Taking the ratio of these amplitudes and substituting for o> 0 its value 
from Eq. (13.3), we get 

C/ m =]/ ~Im (13.9) 

We can also obtain this equation if we proceed from the fact that the 
maximum value of the energy of the electric field y C Um must equal 

the maximum value of the energy of the magnetic field LI\ 


13.3. Free Damped Oscillations 

Any real circuit has a resistance. The energy stored in the circuit 
is gradually spent in this resistance for heating, owing to which the 
free oscillations become damped. Equation 
(5.27) written for circuit 1-3-2 shown in Fig. 13.3 
has the form 

1R= — ^ — L (13.10) 

[compare with Eq. (13.1)1. Dividing this equa- 

• • • 

tion by L and substituting q for I and q for 
dl/dt, we obtain 

'i+irh+Tc^ 0 ( 1311 > 

Taking into account that the reciprocal of LC equals the square of 
the natural frequency of the circuit o> 0 [see Eq. (13.3)1, and introduc- 
ing the symbol 

v=4r < 1S12 > 

Eq. (13.11) can be written in the form 

q + 2p? + w\q — 0 (13.13) 

This equation coincides with the differential equation of damped 
mechanical oscillations [see Eq. (7.11) of Vol. I, p. 1891. 

When < <al, i.e. 7? a /4L a < i/LC , the solution of Eq. (13.13) 
has the form 

? = ?m,o ex P( — P<) (“< + «) (13.14) 


C 




264 


Electricity and Magnetism 


where a> = V ©J — P®. Substituting for oo 0 its value from Eq. (13.3) 
and for p its value from Eq. (13.12), we find that 

01 = V ~LC 4L r (13.15) 

Thus, the frequency of damped oscillations <■» is smaller than the 
natural frequency © 0 . When R = 0, Eq. (13.13) transforms into 
Eq. (13.3). 

Dividing Eq. (13.14) by the capacitance C, we get the voltage 
across the capacitor: 

U — — e -t* cos (©£ -f- a) = f/ m> 0 e -pt cos (<at -|- a) (13.16) 

To find the current, we shall differentiate Eq. (13.14) with respect 
to time 

/ = q = q m , o e ~ W I — P cos (©< + oc) — <D sin (a>t a)) 

Multiplying the right-hand side of this equation by the expression 

<Qq 

equal to unity, we get 

/ = O*- 11 ' [ - CO5(<0l+«)_ 

- yJ+y Sinit + a)] 

Introducing the angle ip determined by the conditions 

n ^ p P_ a{n CO <■) 

cos tp y r ©*-(-p* “ — too smip— ^___ i — 

we can write 

I = <t>o9m. o<?~ p, cos(G>« + a + i|>) (13.17) 

Since cos ip < 0 and sin ip >► 0, the value of ip is within the limits 
from n/2 to n (i.e. n/2 < ip < jt). Thus, when a circuit contains a 
resistance, the current leads the voltage across the capacitor in phase 
by more than n/2 (when R — 0, the advance in phase is n/2). 

A plot of function (13.14) is depicted in Fig. 13.4. Plots of the volt- 
age and current are similar to it. 

It is customary practice to characterize the damping of oscil- 
lations by the logarithmic decrement 

x=ln ^TTT=P 7, < 1318 > 



Electrical Oscillations 


265 


[see Eq. (7.104) of Vol. I, p. 212], Here A (*) is the amplitude of 
the relevant quantity (g, Z7, or /). We remind our reader that the 
logarithmic decrement is the reciprocal of the number of oscilla- 
tions N e performed during the time needed for the amplitude to de- 
crease to i/e of its initial value: 


Using in Eq. (13.18) the value of 0 
from Eq. (13.12) and substituting 2rc/(0 
for T y we get the following expression 
for X: 

l =TET-7 = T§ 

The frequency < 0 , and, therefore, also A, are determined by the para- 
meters of a circuit L, C, and JR. Thus, the logarithmic decrement is 
a characteristic of a circuit. 

If the damping is not great (p a C <oJ), we can assume in Eq. (13.19) 
that a) « © 0 = 1/ Y LC. Hence, 

’k^ - n *y LC =nR \/ £ (13.20) 

An oscillatory circuit is often characterized by its quality, or 
simply Q , determined as a quantity that is inversely proportional to 
the logarithmic decrement: 

Q = ^- = nN e (13.21) 

It follows from Eq. (13.21) that the quality of a circuit is the higher, 
the greater is the number of oscillations completed before the am- 
plitude diminishes to i/e of its initial value. 

For weak damping, we have 

Q = \V (13-22) 

[see Eq. (13.20)1. 

In Sec. 7. 10 of Vol. I , p. 210 et seq. , we showed that when the damp- 
ing is weak, the quality of a mechanical oscillatory system equals 
the ratio of the energy stored in the system at a given moment to the 
decrement of this energy during one period of oscillations with an 
accuracy to the factor 2n. We shall show that this also holds for elec- 
trical oscillations. The amplitude of the current in a circuit dimin- 
ishes according to the law e~ P*. The energy W stored in the circuit 
is proportional to the square of the current amplitude (or to the square 
of the amplitude of the voltage across the capacitor). Hence, W 
diminishes according to the law e The relative reduction in the 





266 


Electricity and Magnetism 


energy during a period is 

AW W(t)—W(t+T ) 1— e-W 4 

W ~ W(t) ~ 1 ~ 1 e 

With insignificant damping (i.e. when X <C 1), we may assume that 
e -2x is approximately equal to i — 2X: 

=1 — (1 — 2X) = 2X 

Finally, substituting the quality Q of the circuit for X in this expres- 
sion in accordance with Eq. (13.21) and solving the equation obtained 
relative to Q, we get 

Q = 2n^ (13.23) 

We shall note in conclusion that when RV4L? l/LC, i.e. when 
p* coj, an aperiodic discharge of the capacitor occurs instead of 
oscillations. The resistance of a circuit at which an oscillatory proc- 
ess transforms into an aperiodic one is called critical. The value of 
the critical resistance R e r is determined by the condition R* T /4L* — 
= MLC, whence 

*cr = 2j/ (13.24) 


13.4. Forced Electrical Oscillations 

To produce forced oscillations of a system, an external periodically 
changing action must be exerted on it. This can be achieved for elec- , 

trical oscillations if we connect a varying 
e.m.f. i n ser ies with t he circuit elements 
or, if after breaking the circuit, we feed> 
an alternating voltage to the contacts 
formed, i.e. the voltage 

U = U m cos <t > t (13.25) 

(Fig. 13.5). This voltage must be added 
to the self-induced e.m.f. As a result, Eq. (13.10) acquires the form 

IR= —^-L~ + U m cosis>t (13.26) 

After transformations, we get the equation 

q + 2$q -f- cos <ot (13.27) 

Here <a* and 8 are determined bv Eos. il3.3> and H3.121. 



O'VrO- 

u 


Fig. 13.5 



Electrical Oscillations 


267 


Equation (13.27) coincides with the differential equation of forced 
mechanical oscillations [see Eq. (7.111) of Vol. I, p. 215], A partial 
solution of this equation has the form 


where 


q = q m cos (at — yp) 


(13.28) 


„ Um/L 

qm VM— a> a )* + 4p*o)* * 


tan yp = 


2pcd 
0)J — co # 


[see Eq. (7.119) of Vol. I, p. 216]. Substitution of their values for 
to % 0 and p gives 




Vm 

to Y fl* + ((oL — l/coC)> 
R 


tan yp 


i/toC — toL 


(13.29) 

(13.30) 


A general solution is obtained if we add the general solution of 
the relevant homogeneous equation to partial solution (13.28). This 
solution was obtained in the preceding section [see Eq. (13.14)]. 
It contains the exponential factor therefore, after sufficient 

time elapses, becomes very small and it may be disregarded. Con- 
sequently, stationary forced oscillations are described by the func- 
tion (13.28). 

Time differentiation of Eq. (13.28) gives the current in a circuit 
with stationary oscillations: 

/= — toq m sin (cot — = cos (tot — + 


(7 m = <o? m ). Let us write this expression in the form* 

/ = I m cos (tot — qp) (13.31) 


where <p = yp — ji/2 is the shift in phase between the current and the 
applied voltage [see Eq. (13.25)]. In accordance with Eq. (13.30): 


tan<p = tanfo-T-) = ~%^= ~ 7T^ < 13 32 > 


Inspection of this equation shows that the current lags in phase be- 
hind the voltage ((p >0) when toL >1 /toC y and leads the voltage 
(<p < 0) when toL <Z l/(oC. According to Eq. (13.29): 


Im Y + — 1/toC)* 


(13.33) 


Let us write Eq. (13.26) in the form 

IR + -1- + L ^ f = U m cos<ot (13.34) 


* We shall not encounter the concept of potential any more up to the end of 
this chapter. Therefore, no misunderstandings will appear if we use the symbol 
q> for the phase angle. 


268 


Electricity and Magnetism 


The product IR equals the voltage U R across the resistance, q!C is 
the voltage across the capacitor U c , and the expression L ( dlldt ) de- 
termines the voltage across the inductance U L . Taking this into 
account, we can write 

U R + U c + U l = U m cos c ot (13.35) 

Thus, the sum of the voltages across the separate elements of a cir- 
cuit at each moment of time equals the voltage applied from an ex- 
ternal source (see Fig. 13.5). 

According to Eq. (13.31) 

U R = RI m cos (<o* — <p) (13.36) 

Dividing Eq. (13.28) by the capacitance, we get the voltage across 
the capacitor 


Here 


U c ^ cos (tot — ty) — U Ct m cos ^(ot — <p 

<?m U m f m 


U c % m — 


«)C \fR 2 + (<oL — 1 /coC) a 


(13.37) 

(13.38) 


[see Eq. (13.33)1. Multiplying the derivative of function (13.31) 
by L, we get the voltage across the inductance: 


U L 



Here 


< oLI m sin (co t—q>) = U Lt m cos (©/ — + 

(13.39) 


U m — <oLI u 


(13.40) 


A comparison of Eqs. (13.31), (13.36), (13.37), and (13.39) shows 
that the voltage across the capacitor lags in phase behind the current 

by ji/2, while the voltage across the 
inductance leads the current by nl 2. The 
voltage across the resistance changes 
in phase with the current. The phase 
relations can be shown very clearly with 
the aid of a vector diagram (see Sec. 7.7 
of Vol. I, p. 203). We remind our read- 
er that a harmonic oscillation (or a 
harmonic function) can be shown with 
the aid of a vector whose length equals 
the amplitude of the oscillation, while 
the direction of the vector makes an 
angle equal to the initial phase of the oscillation with a certain axis. 
Let us take the axis of currents as the straight line from which the 
initial phase is counted. This gives us the diagram shown in Fig. 13.6. 




Electrical Oscillation* 


269 


According to Eq. (13.35), the sum of the three functions U R , U c , and 
Ul must equal the applied voltage U . The voltage U is accordingly 
shown in the diagram by a vector equal to the sum of the vectors 
U R j U c» and f/r,. We must note that Eq. (13.33) is easily obtained 
from the right triangle formed in the vector diagram by the vectors 
U , U R , and the difference U L — U c . 

The resonance frequency for the charge q and the voltage U c 
across the capacitor is 

<*>g f res == <&U, res — — 2{1 2 = -gjT ^ (13.41) 

[see Eq. (7.127) of Vol. I, p. 219]. 

Resonance curves for U c are shown in Fig. 13.7 (resonance curves 
for q have the same form). They are similar to the resonance curves 



Fig. 13.7 Fig. 13.8 


obtained for mechanical oscillations (see Fig. 7.24 of Vol. I, p. 219). 
When g> — 0, the resonance curves converge at one point having the 
ordinate U c% m = U m , i.e. the voltage appearing across the capacitor 
when it is connected to a source of steady voltage U m . The maximum 
in resonance will be- the higher and the sharper, the smaller is P = 
= R/2L y i.e. the smaller is the resistance and the greater the induc- 
tance of the circuit. 

Resonance curves for the current are shown in Fig. 13.8. They 
correspond to the resonance curves for the velocity in mechanical 
oscillations. The amplitude of the current has a maximum value at 
<dL — ifioC = 0 [see Eq. (13.33)]. Consequently, the resonance 
frequency for the current coincides with the natural frequency of the 
circuit <o 0 : 


0 )/. 


(13.42) 



270 


Electricity and Magnetism 


The intercept formed by the resonance curves on the 7 m - axis is 
zero — at a constant voltage, a steady current cannot flow in a cir- 
cuit containing a capacitor. 

At small damping (when |} 2 <C coj), the resonance frequency for 
the voltage can be taken equal to (o 0 [see Eq. (13.41)]. According^ 
we may consider that co rs e£ — l/tOres^ ^ 0. By Eq. (13.38), the 
ratio of the amplitude of the voltage across the capacitor in resonance 
U c% m.res to the amplitude of the external voltage U m will in this 
case be 



(13.43) 


[we have assumed in Eq. (13.38) that — 6>t/, r es — <*> 0 ]. Here Q 
is the quality of the circuit [see Eq. (13.22)]. Thus, the quality of a 

circuit shows how many times the voltage 
across a capacitor can exceed the applied 
voltage. 

The quality of a circuit also determines 
the sharpness of the resonance curves. Fig- 
ure 13.9 shows a resonance curve for the 
current in a circuit. Instead of laying ofl 
the values of 7 m corresponding to a given 
frequency along the axis of ordinates, we 
have laid off the ratio of 7 m to / m ,res 
(i.e. to 7 m in resonance). Let us consider 
the width of the curve Ao> taken at the 
height 0.7 (a power ratio of 0.7 2 « 0.5 
Fig. 13.9 corresponds to a ratio of the current am- 

plitudes equal to 0.7). We can show that 
the ratio of this width to the resonance frequency equals a quantity 
that is the reciprocal of the quality. of a circuit: 



Aco 1 

«>• Q 


(13.44) 


We remind our reader that Eqs. (13.43) and (13.44) hold only for 
large values of Q> i.e. when the damping of the free oscillations in the 
circuit is small. 

The phenomenon of resonance is used to separate the required com- 
ponent from a complex voltage. Assume that the voltage applied to 
a circuit is 


U — U m# | cos (<Dit "j" a,) U | cos ((i> 2 ^ -f" -1- . • • 


By tuning the circuit to one of the frequencies <o lf co*, etc. (i.e. by 
correspondingly choosing its parameters C and L) y we can obtain a 
voltage across the capacitor that exceeds the value of the given com- 


Electrical Oscillations 


271 


ponent Q times, whereas the voltage produced across the capacitor 
by the other components will be weak. Such a process is carried out, 
for example, when tuning a radio receiver to the required wavelength. 


13.5. Alternating Current 


The stationary forced oscillations described in the preceding sec- 
tion can be considered as the flow of an alternating current produced 
by the alternating voltage 

cos cd* (13.45) 


in a circuit including a capacitance, an inductance, and a resistance. 
According to Eqs. (13.31), (13.32), and (13.33), this current varies 
according to the law 

/ = I m cos (<o* — <p) (13.46) 


The amplitude of the current is determined by the amplitude of the 
voltage U my the circuit parameters C, £, /?, and the frequency cd: 




V R* + (<oL — 1/cdC) 2 


(13.47) 


The current lags in phase behind the voltage by the angle q> that 
depends on the parameters of the circuit and on the frequency: 


tan <p = 


o)L — 1 /g>C 
R 


(13.48) 


When <p<0, the current actually leads the voltage. 

The expression 

2 " V «*+(“£-- S)‘ < 13 ' 49 > 

in the denominator of Eq. (13.47) is called the Impedance, 

If a circuit consists only of a resistance i?, the equation of 
Ohm’s law has the form 

IR = U m cos of 

Hence it follows that the current in this case varies in phase with 
the voltage, while the amplitude of the current is 


A comparison of this expression with Eq. (13.47) shows that the 
replacement of a capacitor with 'a shorted circuit section signifies a 
transition to C = oo instead of to C ~ 0. 

Any real circuit has finite values of R y L , and C. It may happen 
that some of these parameters are such that their influence on the 



272 


Electricity and Magnetism 


current may be disregarded. Suppose that R of a circuit may be 
assumed equal to zero, and C equal to infinity. Now, we can see 
from Eqs. (13.47) and (13.48) that 


/ = Um 
Im <&L 


(13.50) 


and that tan 9 » 00 (accordingly, 9 = n/2). The quantity 

X L = d)L (13.51) 

is called the inductive reactance. If L is expressed in henries, and 
co in rad/s, then X L will be expressed in ohms. Examination of 
Eq. (13.51) shows that the inductive reactance grows with the fre- 
quency co. An inductance does not react to a steady current (to = 0), 
i.e. X L = 0. 

The current in an inductance lags behind the voltage by n/2. 
Accordingly, the voltage across the inductance leads the current by 
n/2 (see Fig. 13.6). 

Now let us assume that R and L both equal zero. Hence, according 
to Eqs. (13.47) and (13.48), we have 


tan 9 


/ - 
m i/mC 


00 (i.e» 9 = — n/ 2 ). The quantity 



(13.52) 

(13.53) 


is called the capacitive reactance. If C is expressed in farads, and 
co in rad/s, then Xc will be expressed in ohms. It follows from Eq. 
(13.53) that the capacitive reactance diminishes with increasing 
frequency. For a steady current, X c = 00 — a steady current cannot 
flow through a capacitor. Since 9 = — n/2, the current flowing 
through a capacitor leads the voltage by n/2. Accordingly, the vol- 
tage across a capacitor lags behind the current by n/2 (see Fig. 13.6). 

Finally, suppose that we may assume R to equal zero. In this 
case, Eq. (13.47) becomes 


The quantity 


/ m = 


u m 

| <oL — l/w<7| 


X = <»L—^ = X l -X c 


(13.54) 

(13.55) 


is called the reactance* 

Equations (13.48) and (13.49) can be written in the form 
tan 9 = , Z — VW+X* 



Electrical Oscillation* 


273 


Thus, if the values of the resistance R and the reactance X are laid 
off along the legs of a right triangle, then the length of the hypote- 
nuse will numerically equal Z (see Fig. 13.6). 

Let us find the power liberated in an alternating-current circuit. 
The instantaneous value of the power equals the product of the in- 
stantaneous values of the voltage and current: 

P (t) = U ( t ) I ( t ) = U m cos o )f-/ m cos (cof — <p) (13.56) 

Taking advantage of the formula 

cos a cos p == ■— cos (a — pi + ycos (a + P) 

we can write Eq. (13.56) in the form 

P (t) == j U m I m cos <p + -j U m I m cos (2iot — <p) (13.57) 

Of practical interest is the time-average value P ( t ), which we 
shall denote simply by P. Since the average value of cos (2© t — q>) 
is zero, we have 

p _ SLe^Im. cos q, ( 13 . 58 ) 

Inspection of Eq. (13.57) shows that the instantaneous power fluc- 



tuates about the average value with a frequency double that of 
the current (Fig. 13.10). 

In accordance with Eq. (13.48), 


cos q> = 


R 

Y R' + ^L—ltvC)* ~ 


£ 

z 


(13.59) 


Using this value of cos <p in Eq. (13.58) and taking into account that 
U m /Z = 7 m , we get 


P = 



(13.60) 



274 


Electricity and Magnetism 


The same power is developed by a direct current whose strength is 



(13.61) 


Quantity (13.61) is known as the effective value ol the current. Simi- 
larly, the quantity 


U = 



(13.62) 


is called the effective voltage. 

Expressing the average power through the effective current and 
voltage, we get 

P — UI cos q> (13.63) 


The factor cos <p in this expression is called the power factor. Engi- 
neers try to make cos 9 as high as possible. At a low value of cos 9, 
a large current must be passed through a circuit to obtain the 
required power, and this results in greater losses in the feeder lines. 



PART II 


WAVES 


CHAPTER 14 ELASTIC WAVES 


14.1. Propagation of Waves 
in an Elastic Medium 

If at any place of an elastic (solid or fluid) medium its particles 
are made to oscillate, then owing to interaction between the parti- 
cles, this oscillation will propagate in the medium from particle to 
particle with a certain velocity v. The process of the propagation of 
oscillations in space is called a wave. 

The particles of a medium in which a wave is propagating are not 
made to perform translational motion by the wave, they only oscil- 
late about their equilibrium positions. Depending on the direction of 
oscillations of particles relative to the direction of propagation of 
the wave, longitudinal and transverse waves are distinguished. In the 
former, the particles of the medium oscillate along the direction of 
propagation of the wave. In transverse waves, the particles of the 
medium oscillate in directions at right angles to the direction of 
wave propagation. Elastic transverse waves can appear only in a 
medium having a resistance to shear. Therefore, only longitudinal 
waves can appear in fluids. Both longitudinal and transverse waves 
can appear in a solid. 

Figure 14.1 shows the motion of the particles when a transverse 
wave propagates in a medium. The numbers 7, 2 y etc. designate par- 

I 

tides spaced at a distance of uT, i.e. at the distance travelled by 

the wave during one-fourth of the period of the oscillations performed 
by the particles. At the moment of time taken as zero, the wave 
propagating along the axis from left to right reached particle 7. As a 
result, the particle began to move upward from its equilibrium posi- 
tion, carrying the following particles along. After one-fourth of a 
period, particle 7 reaches its extreme top position; simultaneously, 
particle 2 begins to move from its equilibrium position. After an- 
other fourth of a period elapses, the first particle will pass its equi- 
librium nositinn moving downward, the second narticle will reach 



276 


Waves 


its extreme top position, and the third particle will begin to move 
upward from its equilibrium position. At the moment T , the first 
particle will complete a cycle of oscillation and will be in the same 
state of motion as at the initial moment. The wave by the moment 
7\ having covered the path vT , will reach particle 5. 



Figure 14.2 shows how the particles move when a longitudinal wave 
propagates in a medium. All the reasoning relating to the behaviour 
of particles in a transverse wave can also be related to the given case 
with displacements to the right and left substituted for the upward 
and downward ones. A glance at the figure shows that the propagation 
of a longitudinal wave in a medium is attended by alternating com- 
pensations and dilatations of the particles (the places of compensation 
of the particles are surrounded by a dash line in the figure). They 
move in the direction of wave propagation with the velocity v. 



Elastic Waves 


277 


Figures 14.1 and 14.2 show oscillations of particles whose equilib- 
rium positions are on the x-axis. Actually, not only the particles 
along the x-axis, but the entire collection of particles contained in a 
certain volume oscillate. Spreading from the source of oscillations, 
the wave process involves new and new parts of space. The locus of 
the points reached by the oscillations at the moment of time t is 



called the wavefront. The latter is the surface separating the part 
of space already involved in the wave process from the region in 
which oscillations have not yet appeared. 

The locus of the points oscillating in the same phase is known as 
a wave surface. A wave surface can be drawn through any point of 
the space involved in a wave process. Hence, there is an infinitely 
great number of wave surfaces, whereas there i§ only one wavefront 
at each moment of time. Wave surfaces remain stationary (they pass 
through the equilibrium positions of particles oscillating in the 
same phase). A wavefront is in constant motion. 

Wave surfaces can have any shape. In the simplest cases, they are 
planes or spheres. The wave in these cases is called plane or spheri- 
cal, accordingly. In a plane wave, the wave surfaces are a multitude 
of parallel planes, in a spherical wave they are a multitude of con- 
centric spheres. 

Assume that a plane wave is propagating along the x-axis. Hence, 
all the points of the medium whose equilibrium positions have an 
identical coordinate x (but different values of y and z) oscillate in 
the same phase. Figure 14.3 shows a curve that produces the displace- 
ment ^ of points having different x’s at a certain moment of time 
from their equilibrium position. This figure must not be understood 
as a visible image of a wave. It shows a graph of the function £ (x, t) 
for a certain fixed moment of time t . Such a graph can be constructed 
for both a longitudinal and a transverse wave. 

The distance X covered by a wave during the time equal to the pe- 
riod of oscillations of the particles of a medium is called the wave- 
length. It is obvious that 

X = vT (14.1) 

where v = velocity of the wave 
T = period of oscillations. 



278 


Wave 9 


The wavelength can also be defined as the distance between the 
closest points of a medium that oscillate with a phase difference of 
2 ji (see Fig. 14.3). 

Substituting 1/v (v is the frequency of oscillations) for T in Eq. 
(14.1), we get 

Xv = v (14.2) 

We can also arrive at this equation from the following considerations. 
In one second, a wave source completes v oscillations, producing 
during each oscillation one “crest” and one “trough” in the medium. 
By the moment when the source will complete its v-th oscillation, 
the first crest will cover the path v . Consequently, the path v must 
contain v crests and troughs of the wave. 


14.2. Equations of a Plan© 
and a Spherical Wave 


A wave equation is an expression that gives the displacement of 
an oscillating particle as a function of its coordinates x, y, z, and 
the time t: 


l = l (*, y, 2 , *) 


(14.3) 


(we have in mind the coordinates of the equilibrium position of the 
particle). This function must be periodical both relative to the time 

t and to the coordinates x, y, z. Its periodicity 
in time follows from the fact that £ describes 
the oscillations of a particle having the coor- 
dinates x, y, z. Its periodicity with respect 
to the coordinates follows from the fact that 
points at a distance X from one another oscil- 
late in the same way. 

Let us find the form of the function £ for a 
plane wave assuming that the oscillations are 
harmonic. For simplicity, we shall direct the 
coordinate axes so that the x-axis coincides 
with the direction of propagation of the wave. 
The wave surfaces will therefore be perpen- 
dicular to the x-axis and, since all the points of the wave surface 
oscillate identically, the displacement g will depend only on x and 
on t y i.e. £ = 5 fa, t). Let the oscillations of the points in the plane 
x = 0 (Fig. 14.4) have the form 

£ (0, t) = A cos (cof + a) 


X“0 





Fig. 14.4 


Let us find the form of the oscillations of the points in the plane cor- 
responding to an arbitrary value of x. To travel the path from the 



Elastic Waves 


279 


plane z = 0 to this plane, the wave needs the time x = x/v (here v 
is the velocity of wave propagation). Consequently, the oscillations 
of the particles in the plane x will lag in time by x behind the oscil- 
lations of the particles in the plane x = 0, i.e. they will have the 
form 

\ (x, t) = A cos [(a (t — x) + a] = A cos -f aj 

Thus, the equation of a plane wave (both a longitudinal and a 
transverse one) propagating in the direction of the x-axis has the 
following form: 

|«^4cos [to [t—\ (14.4) 

The quantity A is the amplitude of a wave. The initial phase of the 
wave a is determined by our choice of the beginning of counting x 
and t. When considering one wave, the initial time and the coordi- 
nates are usually selected so that a is zero. This cannot be done, as 
a rule, when considering several waves jointly. 

Let us fix a value of the phase in Eq. (14.4) by assuming that 


© [t — -7) +« = const 


(14.5) 


This expression determines the relation between the time t and the 
place x where the phase has a fixed value. The value of dxldt en- 
suing from it gives the velocity with which the given value of the 
phase propagates. Differentiation of Eq. (14.5) yields 


whence 


dt ~dx = 0 

v 



(14.6) 


Thus, the velocity of wave propagation u in Eq. (14.4) is the velo- 
city of phase propagation, and in this connection it is called the 
phase velocity. 

According to Eq. (14.6), we have dxldt >0. Hence, Eq. (14.4) 
describes a wave propagating in the direction of growing x . A wave 
propagating in the opposite direction is described by the equation 

6 — ,4cos[©(* + -^-)+a] (14.7) 

Indeed, equating the phase of wave (14.7) to a constant and differen- 
tiating the equation obtained , we arrive at the expression 

dx 


280 


Waves 


from which it follows that the wave given by Eq. (14.7) propagates 
in the direction of diminishing x . 

The equation of a plane wave can be given a symmetrical form 
relative to x and U For this purpose, let us introduce the quantity 

(14.8) 

known as the wave number. Multiplying the numerator and the de- 
nominator of Eq. (14.8) by the frequency v, we can represent the 
wave number in the form 

(14.9) 

[see Eq. (14.2)]. Opening the parentheses in Eq. (14.4) and taking 
Eq. (14.9) into account, we arrive at the following equation for a 
plane wave propagating along the x-axis: 

| = A cos (c ot — kx + a) (14*10) 

The equation of a wave propagating in the direction of diminishing x 
differs from Eq. (14.10) only in the sign of the term kx. 

In deriving Eq. (14.10), we assumed that the amplitude of the 
oscillations does not depend on x. This is observed for a plane wave 
when the energy of the wave is not absorbed by the medium. When 
a wave propagates in a medium absorbing energy, the intensity of 
the wave gradually diminishes with an increasing distance from the 
source of oscillations — damping of the wave is observed. Experi- 
ments show that in a homogeneous medium such damping occurs 
according to an exponential law: A = A 0 e~v x [compare with the 
diminishing of the amplitude of damped oscillations with time; 
see Eq. (7.102) of Vol. I, p. 210]. Accordingly, the equation of a 
plane wave has the following form: 

£ = A 0 e-v*cos (<of — kx + a) (14.11) 

(A 0 is the amplitude at points in the plane x = 0, and y is the at- 
tenuation coefficient). 

Now let us find the equation of a spherical wave. Any real source 
of waves has a certain extent. But if we limit ourselves to consider- 
ing a wave at distances from its source appreciably exceeding the 
dimensions of the source, then the latter may be treated as a point 
one. A wave emitted by a point source in an isotropic and homoge- 
neous medium will be spherical. Assume that the phase of oscillations 
of the source is (<o£ + a)- Hence, points on a wave surface of radius 
r will oscillate with the phase o> (t — r/v) + a = at — kr + a 
(the wave needs the time t = r/v to travel the path r). The ampli- 
tude of the oscillations in this case, even if the energy of the wave 
is not absorbed by the medium, does not remain constant — it di- 




Elastic Waves 


281 


minishes with the distance from the source as 1/r (see Sec. 14.6). 
Consequently, the equation of a spherical wave has the form 

£ = -£-cos (cof — fcr + a) (14.12) 

where A is a constant quantity numerically equal to the amplitude 
at a distance of unity from the source. The dimension of A equals 
that of the oscillating quantity multiplied by the dimension of 
length. The factor e ~ Yr must be added to Eq. (14.12) for an ab- 
sorbing medium. 

We remind our reader that owing to the assumptions we have made 
Eq. (14.12) holds only when r appreciably exceeds the dimensions 
of the source. When r tends to zero, the expression for the amplitude 
tends to infinity. The explanation of this absurd result is that the 
equation cannot be used for small r’s. 


14.3. Equation of a Plane Wave Propagating 
in an Arbitrary Direction 

Let us find the equation of a plane wave propagating in a direction 
making the angles a, p, y with the coordinate axes x, y , z. We shall 
assume that the oscillations in a plane passing through the origin 
of coordinates (Fig. 14.5) have the form 

l 0 = A cos (at + «) (14.13) 

Let us take a wave surface (plane) at the 
distance l from the origin of coordinates. 

The oscillations in this plane will lag 
behind those expressed by Eq. (14.13) by 
the time t = l/v: 

£= ,4 COS [(0 +ClJ = 

= A cos (at — kl + a) (14.14) 

(ft == a/v; see Eq. (14.9)]. 

Let us express l through the position 
vector of points on the surface being con- 
sidered. For this purpose, we shall introduce the unit vector n of a 
normal to the wave surface. A glance at Fig. 14.5 shows that the sca- 
lar product of n and the position vector r of any point on the surface 
is l: 

nr = r cos <p = l 

Substitution of nr for / in Eq. (14.14) yields 

l = A cos (o)f — knr + a) 



(14.15) 


282 


Waves 


The vector 

k = kn (14.16) 

equal in magnitude to the wave number k = 2n/X and directed along 
a normal to the wave surface is called the wave vector. Thus, 
Eq. (14.15) can be written in the form 

£ (r, t) = A cos (tot — kr + a) (14.17) 

We have obtained the equation for a plane undamped wave propagat- 
ing in the direction determined by the wave vector k. For a damped 
wave, the factor = e“ Ynr must be added to the equation. 

Function (14.17) gives the deviation of a point having the position 
vector r from its equilibrium position at the moment of time t (we 
remind our reader that r determines the equilibrium position of the 
point). To pass over from the position vector of a point to its coor- 
dinates x, j/y Zy let us express tli e scalar product kr through the com- 
ponents of the vectors along the coordinate axes: 

kr = k^x + k v y + k z z 

The equation of a plane wave therefore becomes 

£ (x, j/y z; t) = A cos (c ot — kyX — k y y — k z z + a) (14.18) 

Here 

k x =-^- cos a, Ar„ = -^cosp, Jfe^-^-cosv (14.19) 

Function (14.18) gives the deviation of a point having the coordi- 
nates x, y, z at the moment of time t . When n coincides with e x , we 
have k x = x, k y = k z = 0, and Eq. (14.18) transforms into Eq. 
(14.10). It is very convenient to write the equation of a plane wave 
in the form 

l = Re {AeW-*'**)) (14.20) 

The symbol Re is usually omitted, having in mind that only the real 
part of the relevant expression is taken. In addition, the complex 
number 

A = Ae*« (14.21) 

called the complex amplitude is introduced. The magnitude of this 
number gives the amplitude, and the argument, the initial phase of 
the wave. 

Thus, the equation of a plane undamped wave can be written in 
the form 

Z=Ae*«»*- k O (14.22) 

The advantages of writing the equation in this form will come to 
light on a later page. 


Elastic Waves 


283 


14.4. The Wave Equation 


The equation of any wave is the solution of a differential equation 
called the wave equation. To establish the form of the wave equation, 
let us compare the second partial derivatives with respect to the coor- 
dinates and time of function (14.18) describing a plane wave. Differ- 
entiating this function twice with respect to each of the variables, 
we get 

cos (cot — kr + a) = — ' o 2 £ 
k\A cos (tot — kr + a) = — k% £ 
k\A cos (cd£ — kr + a ) = — 
k\A cos (a)* — kr + a) = — k\^ 


d 2 l 
dx 2 

<n 

dy 2 
dz* 


Summation of the derivatives with respect to the coordinates yields 


d*t . 


+ -3-=-(*i + *S + *I)&=-* 2 S 


~j)x* -* dy* ” 1 -r "V -r «•*/ b «• b (14.23) 

Comparing this sum with the time derivative and substituting 1/v * 
for A 2 /© 2 (see Eq. (14.9)], we get the equation 

0*S , 0*1 , d% _ i a*t 


+ 


dx 2 1 dy 2 ' dz 2 v 2 dt 2 

This is exactly the wave equation. It can be written in the form 


(14.24) 


A£ = 


1 d*l 
v* dt* 


(14.25) 


where A is the Laplacian operator (see Eq. (1.104)]. 

It is easy to convince ourselves that the wave equation is satisfied 
not only by function (14.18), but also by any function of the form 

/ (x, y, z; t) — f (at — A*x — k v y — k z z + a) (14.26) 


Indeed, denoting the expression in parentheses in the 
side of Eq. (14.26) by £, we have 


Similarly 


dt d£ dt 


= /'co, 


d 2 f df' 
dt 2 _(D d; dt 


(0 2 f 





right-hand 

(14.27) 

(14.28) 


Introducing Eqs. (14.27) and (14.28) into Eq. (14.24), we arrive 
at the conclusion that function (14.26) satisfies the wave equation if 
we assume that v — <o Ik. 



284 


Waves 


Any function satisfying an equation of the form of Eq. (14.24) 
describes a wave; the square root of the quantity that is the recipro- 
cal of the coefficient of d*l,/dt 2 gives the phase velocity of this wave. 
We must note that for a plane wave propagating along the x-axis t 
the wave equation has the form 


_ 1 


ax* 


v 3 a<* 


(14,29) 


14.5. Velocity of Elastic Waves 
in a Solid Medium 




Assume that a longitudinal plane wave propagates in the direction 
of the x-axis. Let us separate in the medium a cylindrical volume 

with a base area of S and a height of Ax 
(Fig. 14.6). The displacements £ of par- 
ticles with different x’s are different at 
each moment of time (see Fig. 14.3 
showing £ against x). If the base of the 
cylinder with the coordinate x has at a 
certain moment of time the displace- 
ment then the displacement of a base 
' I — with the coordinate x + Ax will be 
£ + A£. Therefore, the volume being 
considered will be deformed — it re- 
ceives the elongation A£ (A^ is an alge- 
braic quantity, A£<0 corresponds to 
compression of the cylinder) or the rel- 
^ ative elongation AS/ Ax. The quantity 

AS/Ax gives the average deformation of 
the cylinder. Since S varies with x ac- 
Fig. 14.6 cording to a non-linear law, the true 

deformation in different cross sections of the cylinder will differ. To 
obtain the deformation (strain) in the cross section x, we must 
make Ax tend to zero. Thu 

iL 

dx 


f rn 
i 

i 

1 

AxJ 

1 

\ 

1 

1 

1 

1 

1 

I 

1 

■ 

1 

1 

1 

V, L_ 

1 

1 

l 

l 

!•-- 


e = - 


(14.30) 


(we have used the symbol of the partial derivative because S depends 
not only on x, but also on t). 

The presence of tensile strain points to the existence of the normal 
stress a which at small strains is proportional to the strain. Ac- 
cording to Eq. (2.30) of Vol. I, p. 67, 

- p- r ^5 (4 A 



Elastic Waves 


285 


( E is Young's modulus of the medium). We must note that the unit 
strain d\!dx and, consequently, the stress a at a fixed moment of 
time depend on x (Fig. 14.7). Where the deviations of the particles 
from their equilibrium position are maximum, the strain and the 
stress are zero. Where the particles are passing through their equili- 
brium position, the strain and stress reach their maximum values, 
the positive and negative strains (i.e. tensions and compressions) 
alternating. Accordingly, as we 
have already noted in Sec. 14.1, 
a longitudinal wave consists of 
alternating compressions and di- 
latations of the medium. 

Let us revert to the cylindrical 
volume depicted in Fig. 14.6 and 
write an equation of motion for 
it. Assuming that Ax is very 
small, we can consider that the 
projection of the acceleration onto 
the x-axis is the same for all Fig. 14*7 

points of the cylinder and is 

d 2 %/dt 2 . The mass of the cylinder is p*S Ax, where p is the density of 
the undeformed medium. The projection onto the x-axis of the force 
acting on the cylinder equals the product of the area S of the cyl- 
inder base and the difference between the normal stresses in the cross 
sections (x + Ax + £ + A£) and (x + £): 



F X = SE 


_! 

r— 

tU>| 

LV dx /x+A 3C+S+&6 ' 

l dx Jx+ZJ 


(14.32) 


The value of the derivative 5£/<9x in the section x 4* 6 can be 
written with great accuracy for small values of 6 in the form 

(#U-(£).+[w(4)].M4),+-&« < 14 - 33 » 

where by <? 2 |/dx 2 is meant the value of the second partial derivative 
of £ with respect to x in the cross section x. 

Owing to the smallness of the quantities Ax, 5, and A|, we can 
perform transformation (14.33) in Eq. (14.32): 

f «= s ML(4l+^ Wx+5+a5) H(#), + 1N}= 

= S£-gf-(Ax + AI)s*.S£-g|-A* < 14 34 > 


(the relative elongation d\ldx in elastic deformations is much smal- 
ler than unity. Consequently, A£ <C Ax so that the addend A£ in 
the sum (Ax + A£) may be disregarded). 


286 


Waves 


Introducing the found values of the mass, acceleration, and force 
into the equation of Newton's second law, we get 


Finally, cancelling S Ax, we arrive at the equation 

d't = p 9*1 
dx * E dt* 


(14.35) 


which is the wave equation for the case when \ is independent of y 
and z. A comparison of Eqs. (14.29) and (14.35) shows that 

v-Vt < 14 ’ 36 > 

Thus, the phase velocity of longitudinal elastic waves equals the 
square root of Young’s modulus divided by the density of the medium. 
Similar calculations for transverse waves lead to the expression 

v=]/ r (14.37) 

where G is the shear modulus. 


14.6. Energy of an Elastic Wave 

Assume that the plane longitudinal wave 

£ = A cos (g>* — kx + a) 

[see Eq. (14.10)1 is propagating in the direction of the x-axis in a 
certain medium. Let us separate in this medium an elementary 
volume AV so small that the velocity and the strain at all the points 
of this volume may be considered the same and equal, respectively, 
to d\!dt and d\!dx. 

The volume we have separated has the kinetic energy 

AW * = -r(lf) %l AV (14.38) 

(p AV is the mass of the volume, and d\!dt is its velocity). 

According to Eq. (3.81) of Vol. I, p. 99, the volume being con- 
sidered also has the potential energy of elastic deformation 



(e = d\/dx is the relative elongation of the cylinder, E is Young’s 
modulus of the medium). Let us use Eq. (14.36) to substitute 
pv* for Young’s modulus (p is the density of the medium, and v is 



Elastic Waves 


287 


the phase velocity of the wave). Hence, the expression for the poten- 
tial energy of the volume AV acquires the form 

= AV (14.39) 

The sum of Eqs. (14.38) and (14.39) gives the total energy 

= AW^k + AW r p = -^-p[(-|^-) 2 + y 2 (-g-) 2 ] AV (14.40) 

Dividing this energy by the volume AF in which it is contained, we 
get the energy density 

“’-4p[(-tr+‘'M4n (“•«> 

Differentiation of Eq. (14.10) once with respect to t and another 
time with respect to x yields 

ot 

— — Acosin (<of — kx + a) 

-J|- = kA sin (<ot — kx + a) 

Introducing these equations into Eq, (14.41) and taking into account 
that k 2 v* = <o 2 , we get 

w = pA 2 o> 2 sin 2 (<of — kx + a) (14.42) 

A similar expression for the energy density is obtained for a trans- 
verse wave. 

It can be seen from Eq. (14.42) that the energy density at each 
moment of time is different at different points of space. At the same 
point, the energy density varies with time as the square of the sine. 
The average value of sine square is one-half. Accordingly, the time- 
averaged value of the energy density at each point of a medium is 

(w) — -j- pA 2 a> a (14.43) 

The energy density given by Eq. (14.42) and its average value [Eq. 
(14.43)] are proportional to the density of the medium p, the square 
of the frequency oa, and the square of the wave amplitude A . Such 
a relation holds not only for an undamped plane wave, but also for 
other kinds of waves (a plane damped wave, a spherical wave, etc.). 

Thus, a medium in which a wave is propagating has an additional 
store of energy. The latter is supplied to the different points of the 
medium from the source of oscillations by the wave itself; conse- 
quently, a wave carries energy with it. The amount of energy carried 
by a wave through a surface in unit time is called the energy flux 



288 


Waves 


through this surface. If the energy dW is carried through a given 
surface during the time dt, then the energy flux <t> is 



(14.44) 


The energy flux is a scalar quantity whose dimension equals that of 
energy divided by the dimension of time, i.e. coincides with the 
dimension of power. Accordingly, O is measured in watts, erg/s, 
etc. 

The energy flux at different points of a medium can have a differ- 
ent intensity. To characterize the flow of energy at different points 

of space, a vector quantity called the density 
of the energy flux is introduced. It numerically 
equals the energy flux through a unit area 
placed at the given point perpendicular to the 

y y direction in which the energy is being trans- 

-»-l vJt U- ferred. The direction of the vector of the energy 

flux density coincides with that of energy 
Fig. 14.8 transfer. 

Assume that the energy AW is transferred 
during the time Af through the area A perpendicular to the direction 
of propagation of a wave. The energy flux density will therefore be 

40 w (14.45) 


’-SF— 


A5 x At 


[see Eg* (14.44)]. The energy A W confined in a cylinder with the base 
ASx and the altitude v A t (v is the phase velocity of the wave) will 
be transferred through the area ASj_ (Fig. 14.8) during the time A t. 
If the dimensions of the cylinder are sufficiently small (as a result 
of the smallness of AS L and Af) to consider that the energy density 
at all points of the cylinder is the same, then AW can be found as 
the product of the energy density w and the volume of the cylinder 
equal to A S±v At: 

AW = wAS ± vAt 


Using this expression in Eq. (14.45), we get the following equation 
for the density of the energy: 

j = wv (14.46) 

Finally, introducing the vector v whose magnitude equals the phase 
velocity of the wave and whose direction coincides with that of 
wave propagation (and energy transfer), we can write 

j = u;v (14.47) 

We have obtained an expression for the vector of the energy flux 
density. This vector was first introduced by the outstanding Russian 
physicist Nikolai Umov (1846-1915) and is called Umov’s vector. 



Elastic Waves 


289 


The vector given by Eq. (14.47), like the energy density u>, is differ- 
ent at different points of space. At a given point, it varies in time 
according to a sine square law. Its average value is 

<j> = (w) V = pA*<a z v (14.48) 



(see Eq. (14.43)1. Equation (14.48), like Eq. (14.43), holds for a 
wave of any kind (spherical, damped, etc.). 

We shall note that when we speak of the inten- 
sity of a wave at a given point, we have in mind 
the time-averaged value of the density of the 
energy flux transferred by the wave. 

Knowing j for all the points of an arbitrary 
surface S , we can calculate the energy flux through 
this surface. For this purpose, let us divide the 
surface into elementary areas dS. During the time 
dt, the energy dW confined in the oblique cylin- 
der shown in Fig. 14.9 will pass through area 
dS. The volume of this cylinder is dV = v dt dS cos 9 . It contains 
the energy dW = w dV = wv dt dS cos 9 (here w is the instan- 
taneous value of the energy density where area dS is). Taking into 
account that 

wv dS cos 9 = f dS cos 9 j dS 


vdt 

Fig. 14.9 


(dS = n dSi see Fig. 14.9), we can write: dW = j dS dt. Hence, 
we obtain the following equation for the energy flux d<t> through 
area dS : 



jdS 


(14.49) 


[compare with Eq. (1.72)]. The total energy flux through a surface 
equals the sum of the elementary fluxes given by Eq. (14.49): 



(14.50) 


We can say in accordance with Eq. (1.74) that the energy flux equals 
the flux of the vector j through surface S. 

Substituting for the vector j in Eq. (14.50) its time-averaged 
value, we get the average value of <P: 

<<D> = f (j) dS (14.51) 


Let us calculate the mean value of the energy flux through an 
arbitrary wave surface of an undamped spherical wave. At each 
point of this surface, the vectors } and dS coincide in direction. In 



290 


Wave$ 


addition, the magnitude of the vector j for all points of the surface 
is identical. Hence, 

(O) = j </> dS = (j) S — <;•> 4itr 2 

(r is the radius of the wave surface). According to Eq. (14.48), we 
have</> = pA 2 <d 2 i>. Thus, 

(O) = 2np<o 2 AJr 2 

(A r is the amplitude of the wave at a distance r from its source). 
Since the energy of the wave is not absorbed by the medium, the 
average energy flux through a sphere of any radius must have the 
same value, i.e. the condition 

A^r 2 = const 

must be observed. It follows that the amplitude A r of an undamped 
spherical wave is inversely proportional to the distance r from the 
wave source [see Eq. (14.12)1. Accordingly, the mean density of the 
energy flux (/) is inversely proportional to the square of the distance 
from the source. 

For a plane damped wave, the amplitude diminishes with the dis- 
tance according to the law A = A 0 e~ yx [see Eq. (14.11)1. The 
average density of the energy flux (i.e. the wave intensity) corre- 
spondingly diminishes according to the law 

7 = 7V-** (14.52) 

Here x = 2y is a quantity called the wave absorption coefficient* 
Its dimension is the reciprocal of that of length. It is easy to see 
that the reciprocal of x equals the distance over which the intensity 
of a wave diminishes to i/e of its initial value. 


14.7. Standing Waves 

If several waves propagate in a medium simultaneously, then 
the oscillations of the particles of the medium will be the geometri- 
cal sum of the oscillations which the particles would perform if each 
of the waves propagated separately. Hence, the waves are simply 
superposed onto one another without disturbing one another. This 
statement following from experiments is called the principle of 
superposition of waves. 

When the oscillations due to separate waves at each point of a me- 
dium have a constant phase difference, the waves are called coherent. 
(A stricter definition of coherence will be given in Sec. 17.2.) The 
summation of coherent waves gives rise to the phenomenon of in- 


Elastic Waves 


291 


terference, consisting in that the oscillations at some points amplify, 
and at other points weaken one another. 

A very important case of interference is observed in the superpo- 
sition of two plane waves having the same amplitude and approach- 
ing each other from opposite directions. The resulting oscillatory 
process is called a standing wave. Standing waves are produced 
when waves are reflected from obstacles. The wave striking an ob- 
stacle and the reflected wave travelling toward it in the opposite 
direction as a result of superposition produce a standing wave. 

Let us write the equations of two plane waves propagating along 
the x-axis in opposite directions: 

= A cos (cot — kx + ctj) 

£ 2 = A cos (vt + hx + aj 

Adding these two equations and transforming the result according 
to the formula for the sum of cosines, we get 

S=S t + £ a = 2A cos i.) cos ( t„t+ (14.53) 

Equation (14.53) is the equation of a standing wave. To simplify 
it, let us choose the beginning of reading x so that the difference 
a t — a x vanishes, and the beginning of reading t so that the sum 
Oj 4- a a vanishes. We shall also substitute for the wave number k 
its value 2 n/X. Equation (14.53) now becomes 

|= ^2Acos2n.-j-Jcos ©f (14.54) 

A glance at Eq. (14.54) shows that at every point of a standing 
wave the oscillations have the same frequency as those of the oppo- 
site waves, the amplitude depending on x: 

amplitude = 1 2 A cos 2 n | 

At the points whose coordinates comply with the condition 

2 n-j- = ±nn (n = 0, 1, 2, ...) (14.55) 

the amplitude of the oscillations reaches its maximum value. These 
points are known as antinodes of the standing wave. We obtain the 
values of the antinode coordinates from Eq. (14.55): 

*antt «■ ± n -J- (» = 0, 1, 2, ...) (14.56) 

It must be borne in mind that an antinode is not a single point, but 
a plane whose points have the value of the coordinate x determined 
by Eq. (14.56). 



292 


Waves 


At the points whose coordinates comply with the condition 

= ± (n +-|- J n (» = 0, 1, 2, . . .) 

the amplitude of the oscillations vanishes. These points are called 
the nodes of the standing wave. The points of the medium at the 
nodes do not oscillate. The coordinates of the nodes have the values 

*node = ± ( n + t)t* ( /l = 0» 1«. 2, ...) (14.57) 

A node, like an antinode, is not a single point, but a plane whose 
points have values of the coordinate x determined by Eq. (14.57). 

Node Node Node Node 
t 

t+i 



Fig. 14.10 


Examination of Eqs. (14.56) and (14.57) shows that the distance 
between adjacent antinodes, like that between adjacent nodes, is 
X/2. The antinodes and nodes are displaced relative to one another 
by a quarter of a wavelength. 

Let us revert to Eq. (14.54). The factor {2A cos 2n changes its 

sign after passing through its zero value. Accordingly, the phase of 
the oscillations at different sides of a node differs by ji. This signifies 
that points at different sides of a node oscillate in counterphase. All 
the points between two adjacent nodes oscillate in phase. Figure 14.10 
contains a number of “instantaneous photographs” of the deviations 
of the points from their equilibrium position. The first of them cor- 
responds to the moment when the deviations reach their greatest 
absolute value. The following “photographs” have been made at 
intervals of one-fourth of a period. The arrows show the velocities 
of the particles. 

Differentiating Eq. (14.54) once with respect to t and once with 
respect to x, we find expressions for the velocity of the particles £ 



Elastic Waves 


293 


and the deformation of the medium e: 

£ == =& — 2<o A cos 2 ji sin <ot (14.58) 

e = = — 2 -4^- A sin 2n cos <of (14.59) 

Equation (14.58) describes a standing wave of velocity, and Eq. 
(14.59) one of deformation. 

Figure 14.11 compares “instantaneous 
photographs” of the displacement, veloc- 
ity, and deformation for the time mo- 
ments 0 and 77 4. Inspection of the graphs 
shows that the nodes and antinodes of the 
velocity coincide with their displacement 
counterparts; the nodes and antinodes of 
the deformation, however, coincide with 
the antinodes and nodes of the displace- 
ment, respectively. When £ and e reach 

their maximum values, £ becomes equal 
to zero, and vice versa. Accordingly, the 
energy of a standing wave transforms 
twice during a period, once completely 
into potential energy mainly concentrat- 
ed near the nodes of the wave (where the 
deformation antinodes are), and once 
completely into kinetic energy mainly 
concentrated near the antinodes of the 
wave (where the antinodes of the veloc- 
ity are). The result is the transition of 
energy from each node to its adjacent antinodes and back. The 
time-averaged energy flux in any cross section of the wave is zero. 



Fig. 14.11 


14.8. Oscillations of a String 

When transverse oscillations are produced in a stretched string 
fastened at both ends, standing waves are set up in it, and there 
must be nodes at the places where the string is fastened. Hence, 
only such oscillations are produced with an appreciable intensity in 
a string when the length of the latter is an integer multiple of half 
their wavelength (Fig. 14.12). This gives the condition 

l = n-$- or — (n — i, 2, 3 ...) (14.60) 



294 


Waves 


(l is the length of the string). The following frequencies correspond 
to the wavelengths given by Eq. (14.60): 

v n = ^~ = -^-n (n= 1, 2, 3, — ) (14.61) 

(v is the phase velocity of the wave determined by the string tension 

and the mass per unit length, i.e. the linear 
density of the string). 

The frequencies v n are called the natural 
frequencies of a string. The natural frequen- 
cies are integral multiples of the frequency 



called the fundamental frequency. 

Fig. 14.12 Harmonic oscillations with frequencies 

according to Eq. (14.61) are called natural 
or normal oscillations. They are also known as harmonics. In the 
general case, the oscillation of a string is a superposition of various 
harmonics. 

The oscillations of a string are remarkable in the respect that 
according to classical notions, we get discrete values of one of the 
quantities characterizing the oscillations (their frequency). Such a 
discrete nature is an exception for classical physics. For quantum 
processes, it is the rule rather than an exception. 

14.9. Sound 

If elastic waves propagating in air have a frequency ranging from 
16 to 20 000 Hz, then upon reaching the human ear, they cause a 
sound to be perceived. Accordingly, elastic waves in any medium 
having a frequency confined within the above limits are called sound 
waves or simply sound. Elastic waves with frequencies below 16 Hz 
are called infrasound, and those with frequencies above 20 000 Hz 
are called ultrasound. The human ear does not hear infra- and ul- 
trasounds. 

People distinguish sounds they hear by pitch, timbre (quality), 
and loudness. A definite physical characteristic of a sound wave 
corresponds to each of these subjective appraisals. 

Any real sound is not a simple harmonic oscillation, but is the 
superposition of harmonic oscillations with a definite set of fre- 
quencies. The collection of frequencies of the oscillations present in 
a given sound is called its acoustic spectrum. If a sound contains 
oscillations of all the frequencies within an interval from v' to v", 
then the spectrum is called continuous. If a sound consists of os- 
cillations having the discrete frequencies v lt v 2 , v 3 , etc., then the 




Elastic Waves 


295 


spectrum is known as a line one. Noises have a continuous acoustic 
spectrum. Oscillations with a line spectrum produce the sensation 
of a sound with a more or less definite pitch. Such a sound is called 
a tone sound, or simply a tone. 

The pitch of a tone is determined by its fundamental (lowest) 
frequency. The relative intensity of the overtones (i.e. of the oscilla- 
tions of the frequencies v 2 , v 3 , etc.) 
determines the timbre, or quality, 
of the sound. The different spectral 
composition of sounds produced by 
various musical instruments makes 
it possible to distinguish by ear, for 
example, a flute from a violin or a 
piano. 

By the intensity of a sound is 
meant the time-averaged value of 
the density of the energy flux car- 
ried by a sound wave. To be audi- 
ble, a wave must have a certain 
minimum intensity known as the 
threshold of hearing. This threshold 
differs somewhat for different per- 
sons and depends quite greatly on the frequency of the sound. The 
human ear is most sensitive to frequencies from 1000 to 4000 Hz. 
In this region of frequencies, the threshold of hearing averages about 
10" 12 W/m 2 . At other frequencies, it is higher (see the bottom curve 
in Fig. 14.13). 

At intensities of the order of 1 to 10 W/m 2 , a wave stops being 
perceived as a sound and produces only a feeling of pain and pressure 
in the ear. The value of the intensity at which this occurs is known 
as the threshold of pain (or the threshold of feeling). The pain threshold, 
like the hearing one, depends on the frequency (see the top curve in 
Fig. 14.13; the data given in this figure relate to the average normal 
hearing). 

The subjectively estimated loudness of a sound grows much more 
slowly than the intensity of the sound waves. When the intensity 
grows in a geometric progression, the loudness grows approximately 
in an arithmetical progression, i.e. linearly. On these grounds, the 
loudness level L is determined as the logarithm of the ratio between 
the intensity of the given sound / and the intensity J 0 taken as the 
initial one: 

L = log -j— (14.62) 

* 0 

The initial intensity / 0 is taken equal to 10“ 12 W/m 2 so that the hear- 
ing threshold at a frequency of the order of 1000 Hz is at the zero 
level (L = 0). 




296 


Waves 


The unit of loudness level L determined by Eq. (14.62) is called 
the bell (B). Generally the decibel (dB), which is one-tenth of a bell, 
is preferred. The value of L in decibels is determined by the equation 

L = 101og-jj- (14.63) 

The ratio of two intensities I x and / 2 can also be expressed in 
decibels: 

Liz — 10 log (14.64) 

This equation can be used to express the reduction in the intensity 
(the damping) of a wave over a certain path in decibels. Thus, for 
example, a damping of 20 dB signifies that the intensity has 
dropped to one-hundredth of its initial value. 

The entire range of intensities at which a wave produces a feel- 
ing of sound in the human ear (from 10" 12 to 10 W/m 2 ) corresponds 
to values of the loudness level from 0 to 130 dB. Table 14.1 gives 
approximate values of the loudness level for selected sounds. 

Table 14.1 


Sound 

Loudness level, dB 

Ticking of a clock 

20 

Whisper at a distance of 1 m 

30 

Quiet conversation 

40 

Speech of a moderate loudness 

60 

Loud speech 

70 

Shout 

80 

Noise of an aircraft engine: 
at a distance of 5 m 

120 

at a distance of 3 m 

130 


The energy which sound waves convey with them is extremely 
small. If we assume, for example, that a glass of water completely 
absorbs the entire energy of a sound wave with a loudness level of 
70 dB falling on it (in this case the amount of energy absorbed per sec- 
ond will be about 2 X 10~ 7 W), then to heat the water from room 
temperature to boiling about ten thousand years will be needed. 

Ultrasonic waves can be produced in the form of directed beams 
like beams of light. Directed ultrasonic beams have found a wide- 
spread application for locating objects and determining the distance 
to them in water. The first to put forward the idea of ultrasonic loca- 
tion was the outstanding French physicist Paul Langevin. He imple- 
mented this idea during the first world war for detecting submarines. 








Elastic Waves 


297 


At present, ultrasonic locators are used for detecting icebergs, fish 
shoals, and the like. 

It is general knowledge that by shouting and determining the time 
that elapses until the echo arrives, i.e. the sound reflected by an 
obstacle — a mountain, forest, the surface of the water in a well, 
etc. — we can find the distance to the obstacle by multiplying half 
of this time by the speed of sound. This principle underlies the loca- 
tor (sonar) mentioned above, and also the ultrasonic echo sounder 
used to measure the depth and determine the relief of the sea bottom. 

Ultrasonic location permits bats to orient themselves very well 
when flying in the dark. A bat periodically emits pulses of an ultra- 
sonic frequency and according to the reflected signals received by 
its ears assesses the distances to surrounding objects with a high 
accuracy. 


14.10. The Velocity of Sound in Gases 

A sound wave in a gas is a sequence of alternating regions of com- 
pression and rarefaction of the gas propagating in space. Hence, the 
pressure at every point of space experiences a periodically changing 
deflection A p from its average value 
p coinciding with the pressure exis- 
ting in the gas when waves are 
absent. Thus, the instantaneous 
value of the pressure at a point of 
space can be written in the form 

p' = p + Ap 

Assume that a wave is propagating 
along the x-axis. Let us consider the 
volume of a gas in the form of a 
cylinder with a base area of S and 
an altitude of Ax (Fig. 14.14), as we 
did in Sec. 14.5 when finding the 
velocity of elastic waves in a solid 
medium. The mass of the gas con- 
fined in this volume is pS Ax, where 
p is the density of the gas undis- 
turbed by the wave. Owing to the 
smallness of Ax, the projection of the acceleration onto the x-axis 
for all the points of the cylinder may be considered the same and 
equal to d 2 \!dt 2 . 

To find the projection onto the x-axis of the force exerted on the 
volume being considered, we must take the product of the cylinder 
base area S and the difference between the pressures in the cross 



V. I 1 L. 

t t 


x x+Ax 
Fig. 14.14 


298 


Waves 


sections (x + \) and (x + Ax + i -f A|). Repeating the reasoning 
that led us to Eq. (14.34), we get 

/■,= -%-SAx 

(we remind our reader that when deriving Eq. (14.34) we took advan- 
tage of the assumption A£ Ax]. 

Thus, we have found the mass of the separated volume of gas, 
its acceleration, and the force exerted on it. Now let us write the 
equation of Newton’s second law for this volume of gas: 

<pSA*>-gt.= -gSA* 


After cancelling S Ax, we get 



dp' 

dx 


(14.65> 


The differential equation we have obtained contains two unknown 
functions, namely, £ and p . Let us express one of them through the 
other. To do this, we shall find the relation between the pressure of 
a gas and the relative change in its volume d\!dx . This relation de- 
pends on the nature of the process of compression (or rarefaction) of 
the gas. The compressions and rarefactions of a gas in a sound wave 
follow one another so frequently that adjacent portions of the medi- 
um do not manage to exchange heat, and the process can be consid- 
ered as an adiabatic one. In an adiabatic process, the pressure and 
volume of a given mass of a gas are related by the equation 

pV y = const (14.66) 


where y is the ratio between the heat capacities of the gas at con- 
stant pressure and at constant volume [see Eq. (10.42) of Vol. I t 
p. 2851. 

In accordance with Eq. (14.66): 


p (<SAx) y = p’ [5 (A* + A£)] v = . 

= p> [ 5 ( Ax + -g- Ax ) J * = p’ (SAx) y ( 1 + -ft ) y 

Cancelling (SAx)y yields 

p=p , ( i +- 3T 

Taking advantage of the assumption (d\ldx) <1, let us expand 
the expression (1 + <?£/dx) v into a series by powers of d\ldx and dis- 
regard the terms of the higher orders of smallness. The result is 

p=p'( i + v-Jr) 



Elastic Waves 


299 


Let us solve this equation with respect to p 




(14.67) 


[we have used the formula 1/(1 + x) ^ 1 — x holding for z<l]. 
It is a simple matter to obtain an expression for A p from the rela- 
tion we have found: 


a / dE 

a p=p -p=—yp-£' 


(14.68) 


Since the order of magnitude of y is near unity, it follows from 
Eq. (14.68) that | d\!dx | ^ | A pip |. Thus, the condition d\!dx <C 
<C 1 signifies that the deviation of the pressure from its average val- 
ue is much smaller than the pressure itself. This is indeed true: 
for the loudest sounds, the amplitude of oscillations of the air pres- 
sure does not exceed 1 mmHg, whereas the atmospheric pressure p 
has a value of the order of 10 3 mmHg. 

Differentiating Eq. (14.67) with respect to x, we find that 


dp’ 

dx 



Finally, using this value of dp'/dx in Eq. (14.65), we get the differ- 
ential equation 

p 

dx 2 yp dt 2 

Comparing it with wave equation (14.29), we get the following ex- 
pression for the velocity of sound waves in a gas: 

v=\/ r y-^ (14.69) 

(we remind our reader that p and p are the pressure and the density 
of the gas undisturbed by a wave). 

At atmospheric pressure and conventional temperatures, most gas- 
es are close in their properties to an ideal gas. Therefore, we can 
assume that the ratio p/p for them equals RTIM, where R is the 
molar gas constant, T is the absolute temperature, and M is the 
mass of a mole of a gas [see Eq. (10.22) of Vol. I, p. 280]. Introducing 
this value into Eq. (14.69), we get the following equation for the 
velocity of sound in a gas: 

v =y rj w- (i4 - 70) 


Examination of this equation shows that the velocity of sound is 
proportional to the square root of the temperature and does not de- 
pend on the pressure. 



300 


Waves 


The average velocity of thermal motion of gas molecules is deter- 
mined by the formula 

<Wmol >“l/" -^T 


[see Eq. (11.70) of Vol. I, p. 323]. A comparison of this equation 
with Eq. (14.70) shows that the velocity of sound in a gas is related 
to the average velocity of thermal motion of its molecules by the 
expression 

V = (v m0l ) ]/ -EL (14.71) 


Substitution for y of its value for air equal to 1.4 yields the expres- 

3 

sion v « -j- <i> mo i). The maximum possible value of y is 5/3. In this 

4 

case, v « -g- ( v mo \ >. Thus, the velocity of sound in a gas is of the 

same order of magnitude as the average velocity of thermal motion 
of the molecules, but is always somewhat lower than <u mo i>. 

Let us calculate the value of the velocity of sound in air at a tem- 
perature of 290 K (room temperature). For air, we have y = 1.40, 
and M = 29 X 10” 8 kg/mol. The molar gas constant is R = 
= 8.31 J/(mol-K). Introducing these values into Eq. (14.70), we get 


v 



1.40 x 8.31x290 
29 X 10-* 


340 m/s 


The value of the sound velocity in air which we have found agrees 
quite well with the value found experimentally. 

Let us find the relation between the intensity of a sound wave I 
and the amplitude of the pressure oscillations (Ap) m . We mentioned 
in Sec. 14.9 that by the intensity of sound is meant the average 
value of the density of the energy flux. Hence, 

/ = pA 2 <* 2 v (14.72) 

[see Eq. (14.48)]. Here p is the density of the undisturbed gas, A is 
the amplitude of oscillations of the particles of the medium, i.e. 
the amplitude of the oscillations of the displacement co is the 
frequency, and v the phase velocity of the wave. We must note that 
in the given case the particles of the medium are understood to be 
macroscopic (i.e. including a great number of molecules) volumes, 
and not molecules; the linear dimensions of these volumes are much 
smaller than the wavelength. 

Assume that £ changes according to the law £ = A cos (art — 
— kx -f- a). Hence 

•^r== Ak sin (<«>* — Arx + a) = A sin (art — kx + a) 


Elastic Waves 


301 


Introducing this value into Eq. (14.68), we obtain 


Ap — — ypA -^-sin (art — kx + a) = — (Ap) m sin (co$ — kx+a) 


whence 


>1 _ (Ap) m y a 
ypu> 


(14.73) 


Using this expression in Eq. (14.72), we get 


'=4p 


(Ap)S,d* 
2 r y 2 p*a>* 


a> z v = 


(Ap)S> 

2v*p^ 




Taking into account that = ( yRT/M ) 2 , and (p/p) 2 = (AT/Af) 2 
(see Eq. (14.70) and the text preceding it], we can write that 


r (Ap)j| 

~~ 2pi; 


(14.74) 


We can use this equation to calculate the approximate values of the 
amplitude of air pressure oscillations corresponding to the range of 
loudness levels from 0 to 130 dB. These values range from 3 X 10~ 5 Pa 
(i.e. 2 X 10“ 7 mmHg) to 100 Pa (about 1 mmHg). 

Let us assess the amplitude of oscillations of the particles A and 

that of the velocity of the particles (|) m . We shall begin with an as- 
sessment of the quantity A determined by Eq. (10.73). Taking 
into account that’iVo) = X/2 ji, we get the expression 


1 (Ap)m.^ 01 _(A p U _ (14.75) 

K 2nv p p ' ' 


(y » 1.5, consequently, 2ny « 10). At a loudness of 130 dB, the 
ratio (Ap) m /p has a value of the order of 10~ s , while at a loudness of 
60 dB this ratio is about 2 X 10” 7 . The lengths of sound waves in air 
range from 21 m (at v = 16 Hz) to 17 mm (at v = 20 000 Hz). 
Inserting these values into Eq. (14.75), we find that at a loudness 
of 60 dB the amplitude of oscillations of the particles is about 
4 X 10“ 4 mm for the longest waves and about 3 X 10~ 7 mm for 
the shortest ones. At a loudness of 130 dB, the amplitude of oscilla- 
tions for the longest waves is about 2 mm. 

For harmonic oscillations, the amplitude of the velocity (|) m 
equals that of the displacement A multiplied by the cyclic frequency 

©: (Dm = Am. Multiplying Eq. (14.75) by <o, we get 


(Dm _ 1 (Ap)„ 


<AP)„ 


(14.76) 

v y p p ' ’ 

Consequently, at a loudness of 130 dB, the amplitude of the veloci- 
ty is about 340 m/s x 10~ s = 0.34 m/s. At a loudness of 60 dB, the 
amplitude of the velocity will be of the order of 0.1 mm/s. We must 
note that unlike the displacement amplitude, the velocity amplitude 
does not depend on the wavelength. 




302 


Wavta 


14.11. The Doppler Effect for Sound Waves 

Assume that a device sensing the oscillations of the medium, which 
we shall call a receiver, is placed in a fluid at a certain distance from 
the wave source. If the source and the receiver of the waves are sta- 
tionary relative to the medium in which the wave is propagating, 
then the frequency of the oscillations picked up by the receiver will 
equal the frequency v 0 of the oscillations of the source. If the source 
or the receiver or both are moving relative to the medium, then the 
frequency v picked up by the receiver may differ from v 0 . This phe- 



Fig. 14.15 

nomenon is called the Doppler effect. [It is named after the Austrian 
scientist Christian Doppler (1803-1853) who described the effect for 
light waves.] 

Let us assume that the source and the receiver move along the 
straight line joining them. We shall assume the velocity of the source 
v t to be positive if it moves toward the receiver and negative if it 
moves away from the receiver. Similarly, we shall assume the veloc- 
ity of the receiver u T to be positive if the latter moves toward the 
source and negative if it moves away from the source. 

If the source is stationary and oscillates with the frequency v 0 , 
then by the moment when the source will complete its v 0 -th oscilla- 
tion, the “crest” of the wave produced by the first oscillation will 
travel the path v in the medium (v is the velocity of propagation of 
the wave relative to the medium). Hence, the v 0 “crests” and “troughs” 
of the wave produced by the source in one second will cover 
the length v . If the source is moving relative to the medium with 
the velocity v s , then at the moment when the source completes its 
v 0 -th oscillation, the crest produced by the first oscillation will be 
at a distance of v — v s from the source (Fig. 14.15). Hence, the 
length v — v B will contain v 0 crests and troughs of a wave, so that 
the wavelength will be 

X = (14.77) 

The stationary receiver will be passed in one second by the crests 
and troughs accommodated on the length v. If the receiver is moving 



Elastic Waves 


303 


with the velocity i; r , then at the end of a time interval of one second 
it will pick up the trough which at the beginning of this interval was 
at a distance numerically equal to v from its present position. Thus, 
in one second, the receiver will pick up the oscillations corresponding 
to the crests and troughs accommodated on a length numerically 
equal to v + v T (Fig. 14.16) and will oscillate with the frequency 

v=B ^r i 


Substituting for X its value from Eq. (14.77), we get 


v = v 0 


U-f- P r 
»—v s 


(14.78) 


It follows from Eq. (14.78) that upon such motion of the source 
and the receiver when the distance between them diminishes, the 



p oscillations 


Fig. 14.16 


frequency v picked up by the receiver will be greater than that of 
the source v 0 . If the distance between the source and the receiver 
increases, v will be less than v 0 . 

If the directions of the velocities v s and v r do not coincide with 
the straight line passing through the source and the receiver, then 
the projections of the vectors v s and v r onto the direction of this 
straight line must be substituted for v s and v T in Eq. (14.78). 

Inspection of Eq. (14.78) shows that the Doppler effect for sound 
waves is determined by the velocities of the source and the receiver 
relative to the medium in which the sound propagates. The Doppler 
effect is also observed for light waves, but the equation for the change 
in the frequency differs from Eq. (14.78). This is due to the fact that 
no material medium exists for light waves whose oscillations would 
be “light”. Therefore, the velocities of the source and the receiver of 
light relative to the “medium” are deprived of a meaning. For light, 
we can speak only of the relative velocity of the receiver and the 
source. The Doppler effect for light waves depends on the magnitude 
and direction of this velocity. This effect will be considered for 
light waves in Sec. 21.4. 



CHAPTER 15 ELECTROMAGNETIC 

WAVES 


15.1. The Wave Equation for 
an Electromagnetic Field 

We established in Chapter 9 that a varying electric field sets up a 
magnetic one which, generally speaking, is also varying. This va- 
rying magnetic field sets up an electric field, and so on. Thus, if we 
use oscillating charges to produce a varying (alternating) electro- 
magnetic field, then in the space surrounding the charges a sequence 
of mutual transformations of an electric and a magnetic field pro- 
pagating from point to point will appear. This process will be peri- 
odic in both time and space and, consequently, will be a wave. 

We shall show that the existence of electromagnetic waves follows 
from Maxwell’s equations. For a homogeneous, neutral (p = 0), 
non-conducting (j = 0) medium with a constant permittivity e and 
a constant permeability p, we have 

0B <?H dD 0E 

at at • at ee# at 

VB = ppoVH, VD = ee 0 VE 


Consequently, Eqs. (9.5), (7.3), (9.13), and (2.23) can be written as 
follows: 


[VE] = — pp 0 (15.1) 

VH = 0 (15.2) 

[VH]=ee 0 ^- (15.3) 

VE = 0 (15.4) 

Let us take a curl of both sides of Eq. (15.1): 

(V. [VEJ] = — pp<> [v, -|r] (15.5) 


The symbol V denotes differentiation by coordinates. A change in 
the sequence of differentiation with respect to the coordinates and 
time leads to the equation 



Making such a substitution in Eq. (15.5) and introducing the value 
oivfin hv Ea. 115.31 for the curl of H into the equation obtained, we 



Electromagnetic Waves 


305 


have 


[V, IVE]]= — ee 0 mL 0 — 


(15.6) 


According to Eq. (1.107), [v, [VE]] = V (VE) — AE. Because of 
Eq. (15.4), the first term of this expression is__zero. Consequently, 
the left-hand side of Eq. (15.6) is — AE. Thus, omitting the minus 
signs at both sides of the equation, we obtain 

AE = ee 0 {ifio 


According to Eq. (6.15), we have e 0 |x 0 = 1/c*. The equation can 
therefore be written in the form 


AE = 


ep 

c* 


0*E 

at* 


(15.7) 


Expanding the Laplacian operator, we get 

a*E a*E a*E ep a 2 E 
d % x + ay* + dz 2 — c 2 dt 2 


(15.8) 


Taking a curl of both sides of Eq. (15.3) and performing similar 
transformations, we arrive at the equation 


a*H . a*H a*H _ ep a*H 

dx 2 ay* az* c* at* 


(15.9) 


Equations (15.8) and (15.9) are inseparably related to each other 
because they have been obtained from Eqs. (15.1) and (15.3) each 
of which contains both E and H. 

Equations (15.8) and (15.9) are typical wave equations [see Eq. 
(14.24)1. Any function satisfying such an equation describes a wave. 
The square root of the quantity that is the reciprocal of the coeffi- 
cient of the time derivative gives the phase velocity of this wave. 
Hence, Eqs. (15.8) and (15.9) point to the fact that electromagnetic 
fields can exist in the form of electromagnetic waves whose phase 
velocity is 

V= TW (15 ' 10) 

In a vacuum (i.e. when e = p = 1), the velocity of electromagnetic 
waves coincides with that of light in free space c . 


15.2. Plane Electromagnetic Wave 

Let us investigate a plane electromagnetic wave propagating in a 
neutral non-conducting medium with a constant permittivity e and 
permeability p (p = 0, j = 0, e = const, p = const). We shall di- 


308 


Waves 


rect the x-axifr at right angles to the wave surfaces. Hence E and H, 
and, consequently, their components along the coordinate axes will 
not depend on the coordinates y and z . For this reason, Eqs. (9.15)- 
(9.18) can be simplified as follows: 

dE z dH„ 

~dT =^o — 

dE v dH z 

— =~Wo — 


(15.11) 


dB x dH x A 

-aT = Wo^T = 0 


(15.12) 


0 = ee 0 - 


dt 


dE h 


— ee 0 


dt 


■ = ee 0 


= ee 0 


dE z 

dt 

dE x 


dx 


0 


(15.13) 


(15.14) 


Equation (15.14) and the first of /Eqs. (15.13) show that E x can 
depend neither on x nor on t. Equation (15.12) and the first of Eqs. 
(15.11) give the same result for H x . Hence, E x and H x differing from 
zero can be due only to constant homogeneous fields superposed onto 
the electromagnetic field of a wave. The wave field itself cannot 
have components along the x-axis. It thus follows that the vectors 
E and H are perpendicular to the direction of propagation of the 
wave, i.e, that electromagnetic waves are transverse. We shall as- 
sume in the following that the constant fields are absent and that 
E x = H x = 0. 

The last two equations (15.11) and the last two equations (15.13) 
can be combined into two independent groups 


dE y dH z 

-dT=-V** — ’ 


dH z 

dx 


dEy 

~‘ ee °~w 


(15.15) 


dE Z dHy 

~dx dt • 


dH y dE z 

~di~ =ee °~dr 


(15.16) 


The first group of equations relates the components E u and H z , 
and the second group, the components E z and H v . Assume that there 
was initially set up a varying electric field E v directed along the 
y-axis. According to the second of Eqs. (15.15), this field produces 



Electromagnetic Waves 


307 


the magnetic field H z directed along the z- axis. In accordance with 
the first of Eqs. (15.15), the field H z produces the electric field E y t 
and so on. Neither the field E z nor the field H v is produced. Simi- 
larly, if the field E z was produced initially, then according to Eqs. 
(15.16) the field H y will appear that will set up the field E zy etc. 
In this case, the fields E y and H z are not produced. Thus, to describe 
a plane electromagnetic wave, it is sufficient to take one of the sys- 
tems of equations (15.15) or (15.16) and to assume that the com- 
ponents in the other system equal zero. 

Let us take Eqs. (15.15) to describe a wave, assuming that E z = 
= H y = 0. We shall differentiate the first equation with respect to 
y d dH d dH 

x and make the substitution — . Next introducing 

dH 

- from the second equation, we get a wave equation for E v : 

ox v 


d* E y fy^l d*E„ 

dx 1 ~~e r dt* 


(15.17) 


(we have substituted Me 2 for e 0 ^t 0 ). Differentiating the second of 
Eqs. (15.15) with respect to x y we find a wave equation for H z after 
similar transformations: 


d*H z __ eft d*H x 
dx* c * dt % 


(15.18) 


The equations obtained are a particular case of Eqs. (15.8) and 

( 15 - 9 ). 

We remind our reader that E x = E z = 0 and H x = H y — 0, so 
that E y == E and H z — H . We have retained the subscripts y and z 
of E and H to stress the circumstance that the vectors E and H 
are directed along mutually perpendicular axes y and z . 

The simplest solution of Eq. (15.17) is the function 

E y = E m cos (a >t ~ kx + a t ) (15.19) 

The solution of Eq. (15.18) is similar: 

H z = H m cos (at — kx + a 2 ) (15.20) 

In these equations, to is the frequency of the wave, k is the wave 
number equal to to!v y and a x and a 2 are the initial phases of the oscil- 
lations at points with the coordinate x = 0. 

Introducing functions (15.19) and (15.20) into Eqs. (15.15), we get 

kE m sin ( tot—kx + a t ) = m sin (tot — te+otj) 

kH m sin (tot — kx + a ^) = ee 0 c oE m sin (cai — kx -f- a,) 

For these equations to be satisfied, equality of the initial phases a t 
and a 2 is needed. In addition, the following relations must be oh- 



306 


Waves 


served 


k E jq — HHoOiffa, 
ee„(oE m = kH m 


Multiplying these two equations, we find that 

ee 0 £* = (15.21) 

Thus, the oscillations of the electric and magnetic vectors in an elec- 
tromagnetic wave occur with the same phase (a x = a t ), while the 
amplitudes of these vectors are related by the expression 


— •ffmWpo (15.22) 

For a wave propagating in a vacuum, we have 

-g=-=l/ B- = 1^4n x 10 -7 x 4ji x 9 x 10* = 

Hbx re o' 

= Y (4n) 2 X 900 = 120n 377 (15.23) 

In the Gaussian system of units, Eq. (15.22) becomes 


E m Yl = H m Vii 


(15.24) 


Consequently, for a vacuum, 



we have E m = H m (E m is measured in 
cgse units, and H m in cgsm ones). 

Multiplying Eq. (15.19) by the unit 
vector e y of the y-axis (E y e y = E), 
and Eq. (19.20) by the unit vector 
e z of the 2 -axis (H z e z = H), we get 
equations for a plane electromagnetic 
wave in the vector form 

E = E m cos (a )t — kx) 

p vj- ' ' (15.25) 

H = H m cos (a >t — kx) 1 


(we have assumed that a t = a 2 =0). 

Figure 15.1 shows an “instantaneous photograph” of a plane elec- 
tromagnetic wave. A glance at the figure shows that the vectors E 
and H form a right-handed system with the direction of propagation 
of the wave. At a fixed point of space, the vectors E and H vary with 
time according to a harmonic law. They simultaneously grow from 
zero, and next reach their maximum value in one-fourth of a period; 
if E is directed upward, then H is directed to the right (we look along 
the direction of propagation of the wave). In another one-fourth of a 
period, both vectors simultaneously vanish. Next they again reach 
their maximum value, but this time E is directed downward, and H 
to the left. And, finally, upon completion of a period of oscillation, 



Electromagnetic Waves 


309 


the vectors again vanish. Such changes in the vectors E and H occur 
at all points of space, but with a shift in phase determined by the 
distance between the points measured along the ar-axis. 


15.3. Experimental Investigation of 
Electromagnetic Waves 

The first experiments with non-optical electromagnetic waves 
were conducted in 1888 by the German physicist Heinrich Hertz 
(1857-1894). Hertz produced waves with the aid of a vibrator which 
he had invented. The vibrator consisted of two 
rods separated by a spark gap. When a high volt- 
age was fed to the vibrator from an induction coil, 
a spark jumped through the gap. It shorted the v. 
latter, and damped electrical oscillations were set ^ 
up in the vibrator (Fig. 15.2; the chokes shown 4 
in the figure were intended to prevent the high- 
frequency current from branching off into the 
inductor winding). During the time the spark 
burned, a great number of oscillations were com- 
pleted. They produced a train of electromagnetic 
waves whose length was approximately twice that 
of the vibrator. By placing vibrators of various length at the focus 
of a concave parabolic mirror, Hertz obtained directed plane waves 
whose length ranged from 0.6 to 10 metres. 

Hertz also studied the emitted wave with the aid of a half-wave 
vibrator having a small spark gap at its middle. When such a vibrator 
was placed parallel to the electric field strength vector of the wave, 
oscillations of the current and voltage were produced in it. Since 
the length of the vibrator was equal to X/2 , the oscillations in it 
owing to resonance reached such an intensity that they caused small 
sparks to jump across the spark gap. 

Hertz reflected and refracted electromagnetic waves with the aid 
of large metal mirrors and an asphalt prism (over 1 m in size and 
with a mass of 1200 kg). He discovered that both these phenomena 
obey the laws established in optics for light waves. By reflecting a 
running plane wave with the aid of a metal mirror to the opposite 
direction, Hertz obtained a standing wave. The distance between 
the nodes and antinodes of the wave made it possible to find its 
length X. By multiplying X by the frequency of oscillations v of the 
vibrator, the velocity of the electromagnetic waves was determined, 
and it was found to be close to c. By placing a grate of parallel copper 
wires in the path of waves, Hertz discovered that the intensity of the 
waves passing through the grate changes very greatly when the grate 
is rotated about the beam. When the wires forming the grate were per- 




310 


Waves 


pendicular to the vector E, the wave passed through the grate with- 
out any hindrance. When the wires were arranged parallel to E, 
the wave did not pass through the grate. Thus, the transverse nature 
of electromagnetic waves was proved. 

Hertz’s experiments were continued by the Russian physicist 
Pyotr Lebedev (1866-1912), who in 1894 obtained electromagnetic 
waves 6 mm long and studied how they travel in crystals. He de- 
tected double refraction of the waves (see Sec. 19.3). 

In 1896, the Russian inventor Aleksandr Popov (1859-1905) for 
the first time in history transmitted a message over a distance of 
about 250 metres with the aid of electromagnetic waves (the words 
“Heinrich Hertz” were transmitted). This laid the foundation of radio 
engineering. 


15.4. Energy of Electromagnetic Waves 

Electromagnetic waves transfer energy. According to Eq. (14.46), 
the density of the energy flux can be obtained by multiplying the 
energy density by the wave velocity. 

The density of the energy of an electromagnetic field w consists 
of the density of the energy of the electric field [determined by 
Eq. (4.10)1 and that of the energy of the magnetic field [determined 
by Eq. (8.40)]: 

W E +W H = e8 ^ - a 4- — 2 ~ ’ (15.26) 

The vectors E and H at a given point of space vary in the same phase*. 
Therefore, Eq. (15.22) giving the relation between the amplitude 
values of E and H also holds for their instantaneous values. It thus 
follows that the densities of the energy of the electric and magnetic 
fields of a wave are identical at each moment of time: w B = w H . 
We can therefore write that 

w — 2lv e — ee 0 E 2 (15.27) 

Taking advantage of the fact that EY ee 0 = HV p|A 0 , we can 
write Eq. (15.27) in the form 

w=Y ee 0 Hpo EH = ““T EH 

where v is the velocity of an electromagnetic wave [see Eq. (15.10)1. 
Multiplying the expression found for w by the wave velocity v y 
we get the magnitude of the energy flux density vector 

S = wv = EH o. lr (15.28) 

■ (jj 0 \J - J J 

* This holds only for a non-conducting taedium. The phases of E and H 
do not coincide in a conducting medium. 



Electromagnetic Waves 


311 


The vectors E and H are mutually perpendicular and form a right- 
handed system with the direction of propagation of the wave. For 
this reason, the direction of the vector [EHl coincides with that of 
energy transfer, and the magnitude of this vector is EH . Hence, the 
vector of the density of the electromagnetic energy flux can be writ- 
ten as the vector product of E and H: 

S = [EHl (15.29) 

The vector S is known as the Poynting vector. 

By analogy with Eq. (14.50), the flux 0 of 
electromagnetic energy through surface A s can 
be found by integration: 


<t> = j SdA, 






J 


.30) 


[in Eq. (14.50) the surface area was designat- 
ed by the symbol S ; since this symbol is used to 
designate the Poynting vector, we were forced 
to introduce the symbol A s for the surface 
area]. 

Let us consider a portion of a homogeneous 
cylindrical conductor through which a steady 

current is flowing (Fig. 15.3) as an example of applying Eqs. (15.29) 
and (15.30). We shall first consider that extraneous forces are absent 
on this portion of the conductor. Hence, according to Eq. (5.22), the 
following relation is observed at each point of the conductor: 



j = oE = yE 

The steady current is distributed over the cross section of the 
conductor with an identical density j. Hence, the electric field within 
the limits of the portion of the conductor shown in Fig. 15.3 will 
be homogeneous. Let us mentally separate a cylindrical volume of 
radius r and length l inside the conductor. At each point on the side 
surface of this cylinder, the vector H is perpendicular to the vector 
E and is directed tangentially to the surface. The magnitude of H 

is ^ jr [according to Eq. (7.10), we have 2nrH = /nr 2 ]. Thus, the 

Poynting vector given by Eq. (15.29) is directed toward the axis of 
the conductor at each point on the surface and has the magnitude 

S = EH = ~ Ejr . Multiplying S by the side surface area of the 

cylinder A s equal to 2 nrZ, we find that the following flux of electro- 
magnetic energy enters the volume we are considering: 


<t> = SA t = ±-E1r.2nrl = Ej.nrH= Ej-V (15.31) 


where V is the volume of the cylinder. 



312 


Waves 


According to Eq. (6.4), Ej = p/ 2 is the amount of heat liberated 
in unit time per unit volume of the conductor. Consequently, Eq. 
(15.31) indicates that the energy liberated in the form of Lenz- 
Joule heat is supplied to the conductor through its side surface in 
the form of energy of an electromagnetic field. The energy flux gra- 
dually weakens with deeper penetration into the conductor (both 
the Poynting vector and the surface through which the flux passes 
diminish) as a result of absorption of energy and its conversion into 
heat. 

Now let us assume that extraneous forces whose field is homoge- 
neous are exerted within the limits of the portion of the conductor 
we are considering (E* = const). In this case according to Eq. (5.25), 
at each point of the conductor we have 

j = o(E + E*) = -±-(E + E*) 

whence 

E = pj — E* (15.32) 

We shall consider that the extraneous forces on the portion of the 
circuit being considered do not hamper the flow of the current, but 
facilitate it. This signifies that the direction of E* coincides with 
that of j. Let us assume that the relation pj = E* is observed. Hence, 
according to Eq. (15.32), the electrostatic field strength E at each 
point vanishes, and there is no flux of electromagnetic energy through 
the side surface. In this case, heat is liberated at the expense of the 
work of the extraneous forces. 

If the relation E* > pj holds, then, as can be seen from Eq. (15.32), 
the vector E will be directed oppositely to the vector j. In this case, 
the vectors E and S will have directions opposite to those shown in 
Fig. 15.3. Hence, instead of flowing in, electromagnetic energy 
flows out through the side surface of the conductor into the space 
surrounding it. 

In summarizing, we can say that in the closed circuit of a steady 
current, the energy from the sections where extraneous forces act is 
transmitted to other sections of the circuit not along the conductors, 
but through the space surrounding the conductors in the form of a 
flux of electromagnetic energy characterized by the vector S. 


15.5. Momentum of Electromagnetic Field 

An electromagnetic wave absorbed in a body imparts a momentum 
to the body, i.e. exerts a pressure on it. This can be shown by the 
following example. Assume that a plane wave impinges normally 
onto a flat surface of a weakly conducting body with e and p equal 
to unity (Fig. 15.4). The electric field of the wave produces a current 



Electromagnetic Waves 


313 


of density j = crE in the body. The magnetic field of the wave will 
act on the current with a force whose value per unit volume of the 
body can be found by Eq. (6.43): 

F u .v = IjB] = Ho [ jH] 

The direction of this force, as can be seen from Fig. 15.4, coincides 
with the direction of propagation of the wave. 

The momentum 

dK = F u .v dl = \i 0 jH dl (15.33) 

is imparted to a surface layer having a unit 
area and a thickness of dl in unit time (the vec- 
tors j and H are mutually perpendicular). The 
energy absorbed in this layer in unit time is 

dW = JE dl (15.34) 

It is liberated in the form of heat. Fig. 15.4 

The momentum given by Eq. (15.33) and the 
energy (Eq. (15.34)1 are imparted to the layer by the wave. Let us 
take their ratio, omitting the symbol d as superfluous: 

K H 

W ~ E 

Taking into account that \ioff 2 = e 0 2?*, we get 

K w 1 

— = y e 0 fi 0 = — 

It thus follows that an electromagnetic wave carrying the energy W 
has the momentum 

K = ±-W (15.35) 

The same relation between the energy and the momentum holds for 
particles with a zero rest mass [see Eq. (8.57) of Vol. I, p. 247]. 
This is not surprising because according to quantum notions, an 
electromagnetic wave is equivalent to a flux of photons, i.e. particles 
whose mass (we have in mind their rest mass) is zero. 

Examination of Eq. (15.35) shows that the density of the momen- 
tum (i.e. the momentum of unit volume) of an electromagnetic field is 

K u . y = -±-w (15.36) 

The energy density is related to the magnitude of the Poynting vec- 
tor by the expression S = wc. Substituting Sic for w in Eq. (15.36) 
and taking into account that the directions of the vectors K and S 
coincide, we can write that 

K uv = -i-S = -i-[EH] 



(15.37) 


314 


Wave t 


We shall note that when energy ol any kind is transferred, the 
density of the energy flux equals the density of the momentum mul- 
tiplied by c 2 . Let us consider, for example, a collection of particles 
distributed in space with the density n and flying with a velocity v 
identical in magnitude and direction. In this case, the density of 
the momentum 


K u . v = n 


my 


V 1 — v*lc % 


(15.38) 


The particles carry along energy whose density flux j w equals the 
density of the particle flux multiplied by the energy of one particle: 


ixv^nv 


me 


2 


V 1 — v 2 /c 2 


(15.39) 


It follows from Eqs. (15.38) and (15.39) that 


K 


__ 1 . 
u.v — lw 


(15.40) 


Assume that an electromagnetic wave falling normally on a body 
is completely absorbed by the body. Hence, a unit of surface area 
of the body receives in unit time the momentum of the wave enclosed 
in a cylinder with a base area of unity and an altitude of c . Accord- 
ing to Eq. (15.36), this momentum is ( w/c ) c — w. At the same time, 
the momentum imparted to a unit surface area in unit time equals 
the pressure p on the surface. Hence, for an absorbing surface, we 
have p = w. This quantity pulsates with a very high frequency. 
We can therefore measure its time-averaged value in practice. Thus, 

p = (w) (15.41) 

For an ideally reflecting surface, the pressure will be double this 
value. 

The value of the pressure calculated by Eq. (15.41) is very small. 
For example, at a distance of 1 m from a source of light having an 
intensity of a million candelas, the pressure is only about 10“ 7 Pa 
(about 10~ 9 gf/cm 2 ). Pyotr Lebedev succeeded in measuring the pres- 
sure of light. By carrying out experiments requiring great inven- 
tiveness and skill, Lebedev measured the pressure of light on solids 
in 1900, and on gases in 1910. The results of the measurements com- 
pletely agreed with Maxwell’s theory. 


15.6. Dipole Emission 

An oscillating electric dipole is the simplest system emitting 
electromagnetic waves. An example of such a dipole is the system 
formed by a fixed point charge +? and a point charge — q oscillat- 
ing near it (Fig. 15.5). The dipole electric moment of this system 


Electromagnetic Waves 


315 


varies in time according to the law 

p = —qr = —qle cos to* = p m cos a>t (15.42) 

where r = position vector of the charge — q 
l = amplitude of oscillations 
e = unit vector directed along the dipole axis 
Pm ~ gle- 

Acquaintance with such an emitting system is especially impor- 
tant in connection with the fact that many questions of the inter- 



action of radiation with a substance can be explained classically pro- 
ceeding from the notion of atoms as of systems of charges contain- 
ing electrons that are capable of performing harmonic oscillations 
about their equilibrium position. 

Let us consider the radiation of a dipole whose dimensions are 
small in comparison with the wavelength (l X). Such a dipole is 
called elementary. The pattern of the electromagnetic field in direct 
proximity to the dipole is very complicated. It becomes simplified 
quite greatly in the so-called wave zone of the dipole that begins at 
distances r considerably exceeding the wavelength (r X). If a 
wave is propagating in a homogeneous isotropic medium, then its 
wavefront in the wave zone will be spherical (Fig. 15.6). The vectors 
E and H at each point are mutually perpendicular and are perpen- 
dicular to the ray, i.e. to the position vector drawn to the given point 
from the centre of the dipole. 

Let us call sections of the wavefront by planes passing through the 
dipole axis meridians, and by planes perpendicular to the dipole axis 
parallels. We can now say that the vector E at each point of a wave 
zone is directed along a tangent to the meridian, and the vector H 
along a tangent to the parallel. If we look along the ray r, then the 
instantaneous pattern of the wave will be the same as shown in 
Fig. 15.5, the only difference being that the amplitude in motion 
along the ray gradually diminishes. 


316 


Waoet 


At each point, the vectors E and H oscillate according to the law 
cos ( a>t — Ar). The amplitudes E m and H m depend on the distance r 
to the emitter and on the angle 0 between the direction of the posi- 
tion vector r and the dipole axis (see Fig. 15.6). This dependence has 
the following form for a vacuum: 

E m oc H m oc -i- sin 0 

The average value of the density of the energy flux ( S > is propor- 
tional to the product E m H m , consequently, 

( S ) oc JL sin 2 0 (15.43) 

A glance at this expression shows that the wave intensity changes 
along the ray (at 0 = const) in inverse proportion to the square of 
the distance from the emitter. In addition, it depends on the angle 



0. The emission of a dipole is the greatest in directions at right 
angles to its axis (0 = ji/ 2). There is no emission in the directions 
coinciding with the axis (0 = 0 and «). How the intensity depends 
on the angle 0 is shown very illustratively with the aid of a dipole 
directional diagram (Fig. 15.7). This diagram is constructed so that 
the length of the segment it intercepts on a ray conducted from the 
centre of the dipole gives the intensity of emission at the angle 0. 

The corresponding calculations show that the radiant power P 
of a dipole (i.e. the energy emitted in all directions in unit time) is 
proportional to the square of the second time derivative of the di- 
pole moment: 

P oc (15.44) 

According to Eq. (15.42), p 2 = pm<D 4 cos 2 at. Introduction of this 
value into expression (15.44) yields 

P oc COS 2 (I )f (15.45) 

Time averaging of this expression gives 

(P) « Pm© 4 


(15.46) 



Electromagnetic Waves 


317 


Thus, the average radiant power of a dipole is proportional to the 
square of the amplitude of the electric dipole moment and to the 
fourth power of the frequency. Therefore, at a low frequency, the 
emission of electrical systems (for instance, industrial frequency 

alternating current transmission lines) is insignificant. 

• • • • 

According to Eq. (15.42), we have p = — qr = — qa, where a 

is the acceleration of an oscillating charge. Substitution of this 
♦ • 

expression for p in expression (15.44) yield s* 

Pocq 2 & 2 (15.47) 

Expression (15.47) determines the radiant power not only for oscil- 
lations, but also for arbitrary motion of a charge. A charge travel- 
ling with acceleration produces electromagnetic waves, and the ra- 
diated power is proportional to the square of the charge and the 
square of the acceleration. For example, the electrons accelerated in 
a betatron (see Sec. 10.5) lose energy as a result of radiation mainly 
due to centripetal acceleration a n = v 2 /r. According to expression 
(15.47), the amount of energy lost grows greatly with an increasing 
velocity of the electrons in the betatron (in proportion to v 4 ). Hence, 
the possible acceleration of electrons in a betatron is limited to about 
500 MeV (at a velocity corresponding to this value, the losses due to 
radiation become equal to the energy imparted to the electrons by 
the vortex electric field). 

A charge performing harmonic oscillations emits a monochromat- 
ic wave with a frequency equal to that of the charge oscillations. 
If the acceleration a of the charge does not change according to a 
harmonic law, then the radiation consists of a set of waves of differ- 
ent frequencies. 

According to expression (15.47), the intensity vanishes when a = 
= 0. Consequently, an electron travelling with a constant velocity 
does not emit electromagnetic waves. This holds, however, only for 
the case when the velocity of an electron does not exceed the 
speed of light V\ = d\ ejx in the medium in which the electron is 
travelling. When v e \ >i>i, radiation is observed that was discovered 
in 1934 by the Soviet physicists Sergei Vavilov (1891-1951) and Pa- 
vel Cerenkov (born 1904). 


• The constant of proportionality when SI units are used is Y \i 9 /e Q /6nc 2 9 
and when units of the Gaussian system are used is 2/(3c*). 


PART III OPTICS 


CHAPTER 16 GENERAL 


16.1. The Light Wave 

Light is a complicated phenomenon: in some cases it behaves like 
an electromagnetic wave, in others like a stream of special particles 
(photons). In the present volume, we shall treat wave optics, i.e. 
the range of phenomena based on the wave nature of light. The 
collection of phenomena due to the corpuscular (particulate) nature 
of light will be dealt with in Volume III. 

What oscillates in an electromagnetic wave are the vectors E and 
H. Experiments show that the physiological, photochemical, pho- 
toelectrical and other actions of light are due to the oscillations of 
the electric vector. Accordingly, we shall speak in the following of 
the light vector, having in mind the electric field strength vector. 
We shall meanwhile make no mention of the magnetic vector of a 
light wave. 

We shall designate the magnitude of the light vector amplitude, as 
a rule, by the letter A (sometimes by the symbol E m ). Hence, the 
change in space and time of the projection of the light vector onto 
the direction along which it oscillates will be described by the 
equation 

E ' = A cos (cdJ — kr + a) (16.1) 

where k = wave number 

r s distance measured along the direction of propagation of 
the light wave. 

For a plane wave propagating in a non-absorbing medium, A = 
= const, for a spherical wave, A diminishes in proportion to 1/r, 
and so on. 

The ratio of the speed of a light wave in a vacuum to the phase 
velocitv*!/ in a medium is known as the absolute refractive index 



General 


319 


of the medium and is designated by the letter n . Thus, 



(16.2) 


A comparison with Eq. (15.10) shows that n = V For the over- 
whelming majority of transparent substances, p, does not virtually 
differ from unity. We can therefore consider that 

n = VI (16.3) 

Equation (16.3) relates the optical and the electrical properties of 
a substance. It may seem on the face of it that this equation is 
wrong. For example, for water e = 81, whereas n = 1.33. It must 
be borne in mind, however, that the value e = 81 has been obtained 
from electrostatic measurements. A different value of e is obtained 
for fast-varying electric fields, and it depends on the frequency of 
oscillations of the field. This explains the dispersion of light, i.e. the 
dependence of the refractive index (or speed of light) on the fre- 
quency (or wavelength). Using the value of e obtained for the rele- 
vant frequency in Eq. (16.3) leads to the correct value of n. 

The values of the refractive index characterize the optical density 
of the medium. A medium with a greater n is called optically denser 
than one with a smaller n . Conversely, a medium with a lower n is 
called optically less dense than one with a greater n . 

The wavelengths of visible light are within the following limits: 

A 0 = 0.40-0.76pm (4000-7600 A) (16.4) 


These values relate to light waves in a vacuum. The lengths of light 
waves in substances will have other values. For oscillations of fre- 
quency v, the wavelength in a vacuum is A 0 = c/v. In a medium in 
which the phase velocity of a light wave is v = dn , the wavelength 
has the value X = v/v = c/xn = XJn. Thus, the length of a light 
wave in a medium with the refractive index n is related to the wave- 
length in a vacuum by the expression 

71 = -^- (16.5) 


The frequencies of visible light waves are within the limits 

v = (0.39-0.75) x 10 15 Hz (16.6) 

The frequency of the changes in the vector of the energy flux density 
carried by a wave will be still greater (it equals 2v). Neither. our eye 
nor any other receiver of luminous energy can track such frequent 
changes in the energy flux, hence they register the time-averaged 
flux. The magnitude of the time-averaged energy flux density carried 
by a light wave is called the light intensity I at the given point of 
space. The density of the flux of electromagnetic energy is deter- 



320 


Optics 


mined by the Poynting vector S. Hence, 

I — I <S) | = | <[EH J> | (16.7) 

Averaging is performed over the time of “operation” of the instru- 
ment, which, as we have already noted, is much greater than the 
period of oscillations of the wave. The intensity is measured either 
in energy units (for example, in W/m 2 ), or in light units named “lu- 
men per square metre” (see Sec. 16.5). 

According to Eq. (15.22), the magnitudes of the amplitudes of 
the vectors E and H in an electromagnetic wave are related by the 
expression 

Em V ee 0 = Hm V H-Po — Hm V Po 
(we have assumed that p = 1). It thus follows that 

where n is the refractive index of the medium in which the wave 
propagates. Thus, H m is proportional to E m and n : 

H m oc nE m (16.8) 

The magnitude of the average value of the Poynting vector is pro- 
portional to E m H m . We can therefore write that 

I oc nEm = nA z - (16.9) 

(the constant of proportionality is ty^o/iio). Hence, the light inten- 

CM 

sity is proportional to the refractive index of the medium and the 
square of the light wave amplitude. 

We must note that when considering the propagation of light in a 
homogeneous medium, we may assume that the intensity is propor- 
tional to the square of the light wave amplitude 

I oc A 2 (16.10) 

For light passing through the interface between media, however, the 
expression for the intensity, which does not take the factor n into 
account, leads to non-conservation of the light flux. 

The lines along which light energy propagates are called rays. 
The averaged Poynting vector (S) is directed at each point along a 
tangent to a ray. The direction of (S) in isotropic media coincides 
with a normal to the wave surface, i.e. with the direction of the 
wave vector k. Hence, the rays are perpendicular to the wave sur- 
faces. In anisotropic media, a normal to the wave surface generally 
does not coincide with the direction of the Poynting vector so that 
the rays are not orthogonal to the wave surfaces. 



General 


321 


Although light waves are transverse, they usually do not dis- 
play asymmetry relative to a ray. The explanation is that in natural 
light (i.e. in light emitted by conventional sources) there are oscil- 
lations that occur in the most diverse directions perpendicular to a 
ray (Fig. 16.1). The radiation of a luminous body consists of the 
waves emitted by its atoms. The process of radiation in an indi- 
vidual atom continues about 10^® s. During this 
time, a sequence of crests and troughs (or, as is 
said, a wave train) of about three metres in length 
is formed. The atom “dies out”, and then “flares 
up” again after a certain time elapses. Many atoms 
“flare up” at the same time. The wave trains they 
emit are superposed on one another and form the Fi S* 
light wave emitted by the relevant body. The 
plane of oscillations is oriented randomly for each wave train. 
Therefore, the resultant wave contains oscillations of different direc- 
tions with an equal probability. 

In natural light, the oscillations in different directions follow 
one another rapidly and without any order. Light in which the direc- 
tion of the oscillations has been brought into order in some way or 
other is called polarized. If the oscillations of the light vector occur 
only in a single plane passing through a ray, the light is called plane 
(or linearly) polarized. The order may consist in that the vector E 
rotates about a ray while simultaneously pulsating in magnitude. 
The result is that the tip of the vector E describes an ellipse. Such 
light is called elliptically polarized. If the tip of the vector E describes 
a circle, the light is called circularly polarized. 

We shall deal with natural light in Chapters 17 and 18. For this 
reason, we shall display no interest in the direction of the light 
vector oscillations. The ways of obtaining polarized light and its 
properties are considered in Chap. 19, 

16.2. Representation of Harmonic Functions 
Using Exponents 

Let us form the sum of two complex numbers z x = x x + iy t and 
z t =x 2 + iy 2 : 

% = z v + z x a + iy X ) + (*2 + Wi) = 

= (*i + x 2 ) + i (y t + y 2 ) (16.11) 

It can be seen from Eq. (16,11) that the real part of the sum of com- 
plex numbers equals the sum of the real parts of the addends: 

Re {(z t + z 2 )} = Re {zj + Re {z t } (16.12) 




322 Optics 

Let us assume that a complex number is a function of a certain 
parameter, for example, of the time t: 

z(t) = x ( t ) + iy (t) 

Differentiating this function with respect to t, we get 

dz dx , . dy 

~dt ~dt ' 1 It 

It thus follows that the real part of the derivative of z with respect 
to t equals the derivative of the real part of z with respect to t: 

M£H4- Re < z > < 16 - 13 > 

A similar relation holds upon integration of a complex function. 
Indeed, 

| z (t) dt = £ x(t) dt + i j y (t) dt 

whence it can be seen that the real part of the integral of z (0 equals 
the integral of the real part of z ( t ): 

Re { j z (0 dt) = j Re (z (t) dt} (16.14) 

It is evident that relations similar to Eqs. (16.12), (16.13), and 
(16.14) also hold for the imaginary parts of complex functions. 

It follows from the above that when the operations of addition, 
differentiation, and integration are performed with complex functions, 
and also linear combinations of these operations, the real (imaginary) 
part of the result coincides with the result that would be obtained 
when similar operations are performed with the real (imaginary) 
parts of the same functions*. Using the symbol L to denote a linear 
combination of the operations listed above, we can write: 

Re {L (s lf s 2 , ...)} = L (Re {z 4 }, R e{z 2 }, ...) (16.15) 

The property of linear operations we have established makes it 
possible to use the following procedure in calculations: when per- 
forming linear operations with harmonic functions of the form 
A cos ((of — kxX — k v y — k z z + a), we can replace these functions 
with the exponents 

A exp [ i (cot— k x x — k v y — k z z + a) ] = A exp [ i (cof — k x x — k v y — k z z) 1 

(16.16) 

where A = Ae ia is a complex number called the complex amplitude* 
With such representation, we can add functions, differentiate them 


* We must note that this rule cannot be applied to non-linear operations, 
for example, to the multiplication of functions and squaring them. 


General 


323 


with respect to the variables t , x, y, z, and also integrate over these 
variables. In performing the calculations, we must take the real 
part of the result obtained. The expediency of this procedure is 
explained by the fact that calculations with exponents are consid- 
erably simpler than calculations performed with trigonometric 
functions. 

Passing over to representation (16.16), we in essence add to all 
functions of the kind A cos (oof — k^x — k^y — k z z + a) the ad- 
dends iA sin (of — k^x — k y y — k z z -f a). We remind our reader 
that we have used a similar procedure when studying forced oscil- 
lations (see Sec. 7.12 of Vol. I, p. 215 et seq.). 


16.3. Reflection and Refraction 

of a Plane Wave at the Interface 
Between Two Dielectrics 

Assume that a plane electromagnetic wave falls on the plane inter- 
face between two homogeneous and isotropic dielectrics. The dielec- 
tric in which the incident wave is propagating is characterized by 
the permittivity e lt and the second dielectric by the permittivity e a . 
We assume that the permeabilities are unity. Experiments show that 
in this case, apart from the plane refracted wave propagating in the 
second dielectric, a plane reflected wave propagating in the first 
dielectric is produced. 

Let us determine the direction of propagation of the incident 
wave with the aid of the wave vector k, of the reflected wave with the 
aid of the vector k' and, finally, of the refracted wave with the aid 
of the vector k". We shall find how the directions of k' and k" are 
related to the direction of k. We can do this by taking advantage of 
the fact that the following condition must be observed at the inter- 
face between the two dielectrics: 

E ux = (16.17) 

Here E Xs z and E,, x are the tangential components of the electric 
field strength in the first and second medium, respectively. 

In Sec. 2.7, we proved Eq. (16.17) for electrostatic fields [see 
Eq. (2.44)]. It can easily be extended, however, to time-varying 
fields. According to Eq. (9.5), the circulation of E determined by 
Eq. (2.42) for varying fields must be not zero, but equal to the integ- 
ral j (-B)dS taken over the area of the loop shown in Fig. 2.9: 

Etdl=E Ux a— E 2 ' X a + (E b )2b= — j BdS 



324 


Optica 


Since B is finite, in the limit transition b -*• 0 the integral in the 
right-hand side vanishes, and we arrive at condition (2.43), from 
which follows Eq. (2.44). 

Assume that the vector k determining the direction of propagation 
of the incident wave is in the plane of the drawing (Fig. 16.2). The 
direction of a normal to the interface will be characterized by the 
vector n. The plane in which the vectors k and n are is called the 

plane of incidence of the wave. Let us take 
the line of intersection of the plane of inci- 
dence with the interface between the dielec- 
trics as the x-axis. We shall direct the y-axis 
at right angles to the plane of the dielectric 
interface. The z-axis will therefore be perpen- 
dicular to the plane of incidence, while the 
vector t will be directed along the x-axis 
(see Fig. 16.2). 

It is obvious from considerations of sym- 
metry that the vectors k' and k # can only 
be in the plane of incidence (the media are 
homogeneous and isotropic). Indeed, assume 
that the vector k' has deflected from this 
plane “toward us”. There are no grounds, however, to give such a 
deflection priority over an equal deflection “away from us”. Conse- 
quently, the only possible direction of k' is that in the plane of 
incidence. Similar reasoning also holds for the vector k*. 

Let us separate from a naturally falling ray a plane-polarized com- 
ponent in which the direction of oscillations of the vector E makes 
an arbitrary angle with the plane of incidence. The oscillations of 
the vector E in the plane electromagnetic wave propagating in the 
direction of the vector k are described by the function* 

E = E m exp li (at — kr)] = E m exp [i (c at-k^x — k y y)l 



Fig. 16.2 


(with our choice of the coordinate axes, the projection of the vector 
k onto the z-axis is zero, therefore the addend — k z z is absent in the 
exponent). By correspondingly choosing the beginning of reading t , 
we have made the initial phase of the wave equal zero. 

The field strengths in the reflected and refracted waves are deter- 
mined by similar expressions 

E' = Em exp [i (co'f — k' x x — /^y + a')] 

E" — Em exp [i (to”t — k r x X’—kZy + a”)] 


where a 9 and a * are the initial phases of the relevant waves. 


• More exactly, the real part of this function, but we shall say simply 
function for brevity’s sake. 


General 


325 


The resultant field in the first medium is 

E t = E-f E' = E m exp [i (<of — k x x — k y y)] + 

+ Em ©xp [i (o V t — k x x—k' v y + a')] (16.18) 

In the second medium 

E 2 — E' = Em exp { i (<D*t — k x x — k" v y -f- a") ] (16.19) 

According to Eq. (16.17), the tangential components of Eqs. (16.18) 
and (16.19) must be the same at the interface, i.e. when y = 0. We 
thus arrive at the expression 

E m . % ©xp [ i (<of — k x x) 1 + t exp [ i (to* t — k' x x + a' ) ] = 

= E^ t x exp [i (a/* — k x x-\~ a 0 )] (16.20) 

For condition (16.20) to be observed at any t , all the frequencies 
must be the same: 

<d = co'=<d' (16.21) 

To convince ourselves that this is true, let us write Eq. (16.20) in 
the form 

a exp (fcof) + b exp (i&'t) == c exp (i&*t) 

where the coefficients a, fe, and c are independent of t . The equation 
which we have written is equivalent to the following two: 

a cos to* + 6 cos a>'* = c cos a>”t 
a sin <o£ + 6 sin oa'J = c sin c o n t 

The sum of two harmonic functions will also be a harmonic function 
only if the functions being added have the same frequencies. The 
harmonic function obtained as a result of addition will have the 
same frequency as the summated functions. Hence follows Eq. (16.21). 
We have thus arrived at the conclusion that the frequencies of the 
reflected and refracted waves coincide with that of the incident wave. 

For condition (16.20) to be observed at any x, the projections of 
the wave vectors onto the ar-axis must be equal: 

k x = k x = k" x (16.22) 

The angles 0, 0', and 0* shown in Fig. 16.2 are called the angle of 
incidence, the angle of reflection, and the angle of refraction. A 
glance at the figure shows that k x = k sin 0, k x = k' sin 0', k x = 
= A 0 sin 0*. Equation (16.22) can therefore be written in the form 

k sin 0 = k' sin 0' = k" sin*0* 

The vectors k and k' have the same magnitude equal to < olv x ; the 
magnitude of the vector k * equals Hence 

— sin 0 = — sin 0' = — sin 0" 

Vy v x v t 




326 


Optics 


It thus follows that 


e' = e 

sin 0 v 1 

sin 9* ~v^ n * 2 


(16.23) 

(16.24) 


The relations we haVe obtained are obeyed for any plane-polarized 
component of a natural ray. Hence, they also hold for a natural ray 
as a whole. 

Equation (16.23) expresses the law of reflection of light, according 
to which the reflected ray lies in one plane with the incident ray and the 
normal to the point of incidence ; the angle of reflection equals the angle 
of incidence . 

Equation (16.24) expresses the law of refraction of light, according 
to which the refracted ray lies in one plane with the incident ray and 
the normal to the point of incidence ; the ratio of the sine of the angle of 
incidence to the sine of the angle of refraction is constant for given sub- 
stances . 

The quantity n 12 in Eq. (16.24) is known as the relative refractive 
index of the second substance with respect to the first one. Let us 
write this quantity in the form 



(16.25) 


Thus, the relative refractive index of two substances equals the ratio 
of their absolute refractive indices. 

Substituting the ratio njn x for n 12 in Eq. (16.24), we can write 
the law of refraction in the form 


n t sin 0 = n 2 sin 0* 


(16.26) 


Inspection of this equation shows that when light passes from an opti- 
cally denser medium to an optically less dense one, the rays move 
away from a normal to the interface of the media. An increase in the 
angle of incidence 0 is attended by a more rapid growth in the angle 
of refraction 0*, and when the angle 0 reaches the value 

0 cr = arcsin n i2 (16.27) 


the angle 0" becomes equal to n/2. The angle determined by Eq. 
(16.27) is called the critical angle. 

The energy carried by an incident ray is distributed between the 
reflected and the refracted rays. As the angle of incidence grows, 
the intensity of the reflected ray increases, while that of the refract- 
ed ray diminishes and vanishes at the critical angle. At angles of 
incidence within the limits from 0 cr to n/2, the light wave penetrates 
into the second medium to a distance of the order of a wavelength X 
and then returns to the first medium. This phenomenon is called 
total internal reflection. 


General 


327 


Let us find the relations between the amplitudes and phases oi the 
incident, reflected, and refracted waves. For simplicity, we shall lim- 
it ourselves to the normal incidence of a wave onto the interface 
between dielectrics (we remind our reader that the dielectrics are 
assumed to be homogeneous and isotropic). Assume that the oscilla- 
tions of the vector E in the falling wave occur along the direction 
which we shall take as the x-axis. It follows from considerations of 
symmetry that the oscillations of the vectors E' and E' also occur 
along the x-axis. In the given case, the unit vector x coincides with 
the unit vector e x . Therefore, the condition of continuity of the tan- 
gential component of the electric field strength has the form 

E X + E*=ET X (16.28) 

Expression (16.8) obtained for the amplitude values of E and H 
also holds for their instantaneous values: H oc nE . It thus follows 
that the instantaneous value of the energy flux density is propor- 
tional to nE 2 . Thus, the law of energy conservation leads to the 
equation 

^El^niEZ + nzE? (16.29) 

We must note that the quantities E xy E Xy and E x in Eqs. (16.28) 
and (16.29) are the instantaneous values of the projections. 

Introducing E x — E x into Eq. (16.29) instead of E x [see Eq. (16.28)], 
it is easy to see that 

Using this value of E" x in Eq. (16.28), we find that 




(16.31) 


Examination of Eq. (16.30) shows that the projections of the 
vectors E and E" have identical signs at each moment of time. Hence, 
we conclude that the oscillations in the incident wave and in the 
one passing into the second medium occur at the interface in the 
same phase — when a wave passes through the interface there is no 
jump in the phase. 

It can be seen from Eq. (16.31) that when n 2 < n 1? the sign of E' x 
coincides with that of E x . This signifies that the oscillations in the 
incident and reflected waves occur at the interface in the same phase — 
the phase of a wave does not change upon reflection. If n 2 >n x , 
then the sign of E' x is opposite to that of E xy the oscillations in the 
incident and reflected waves occur at the interface in counterphase — 
the phase of the wave changes in a jump by n upon reflection. The 
result obtained also holds upon the inclined falling of a wave at the 
interface between two transparent media. 




328 


Optics 


Thus, when a light wave is reflected from an interface between an 
optically less dense medium and an optically denser one (when 
n i < the phase of oscillations of the light vector changes by n. 
Such a phase change does not occur upon reflection from an inter- 
face between an optically denser medium and an optically less 
dense one (when >n 2 ). 

Equations (16.30) and (16.31) have been obtained for the instan- 
taneous values of the projections of the light vectors. Similar rela- 
tions also hold for the amplitudes of the light vectors: 


ET m 


2 n x 

**l + **2 


"1 1 E* 

**l+**i m 


(16.32) 


These relations make it possible to find the reflection coefficient p 
and the transmission coefficient x of a light wave (for normal inci- 
dence at the interface between two transparent media). Indeed, by 
definition 


/' 

P_ 1 ~~ 


where V is the intensity of the reflected wave, and I is the intensity 
of the incident one. Using in this equation the ratio E' m !E m obtained 
from Eq. (16.32), we arrive at the formula 



**12 1 \ 2 
**12 H - 1 ' 


(16.33) 


Here n 12 = is the refractive index of the second medium rela- 

tive to the first one. 

We get the following expression for the transmission coefficient: 


x 


I m _ n t Em _ / 2_\2 

1 r^Em 12 V *12+1/ 


(16.34) 


We must note that the substitution for n 12 in Eq. (16.33) of its 
reciprocal n tl — l/n 12 does not change the value of p. Hence, the 
coefficient of reflection of the interface between two given media 
has the same value for both directions of propagation of light. 

The index of refraction for glass is close to 1.5. Introducing n lt = 
= 1.5 into Eq. (16.33), we get p = 0.04. Thus, each surface of a 
glass plate reflects (with incidence close to normal) about four per 
cent of the luminous energy falling on it. 


16.4. Luminous Flux 

A real light wave is a superposition of waves with lengths confined 
within the interval AX . The latter is finite even for monochromatic 
(single-coloured) light. In white light, AX covers the entire range of 



General 


329 


electromagnetic waves perceived by the eye, i.e. it ranges from 
0.40 to 0.76 pm. 

The distribution of the energy flux by wavelengths can be cha- 
racterized with the aid of the distribution function 

<PW = -^T* (16.35) 

where d<J> en is the energy flux falling to the wavelengths from X to 
X + dX. Knowing the form of function (16.35), we can calculate the 
energy flux transferred by waves whose lengths are within the finite 
interval from X* to X 2 : 

f’ 

<I>en = } q> (X) dX (16.36) 

The action of light on the eye (the perception of light) depends quite 
greatly on the wavelength. This is easy to understand if we take 



040 045 050 055 060 065 070 075 A. pm 

Fig. 10.3 

into account that electromagnetic waves with X below 0.40 pm 
and above 0.76 pm are not perceived at all by the human eye. The 
sensitivity of an average normal human eye to radiation of various 
wavelengths can be depicted graphically by a curve of relative spec- 
tral sensitivity (Fig. 16.3). The wavelength X is laid off along the 
horizontal axis, and the relative spectral sensitivity V (X) along the 
vertical one. The eye is most sensitive to radiation of the wavelength 
0.555 pm* (the green part of the spectrum). The function V (X) 
for this wavelength is taken equal to unity. The luminous intensity 
estimated visually for other wavelengths is lower, although the 
energy flux is the same. Accordingly, V (X) for these wavelengths is 

• It is interesting to note that this wavelength is represented with the 
greatest intensity in solar radiation. 




330 


Optic 9 


also less than unity. The values of the function V (X) are inversely 
proportional to the values of the energy fluxes producing a visual 
sensation identical in intensity: 

V(Xi) _ n), 

V (X,) (dCDen)i 

For example, V (X) — 0.5 signifies that for obtaining a visual sen* 
sation of the same intensity, light of the given wavelength must have 
a density of the energy flux twice that of light for which V (X) = t. 
Outside of the interval of visible wavelengths, the function V (X) 
is zero. 

The quantity d> called the luminous flux is introduced to character- 
ize the luminous intensity with account of its ability to produce a 
visual sensation. For the interval dX, the luminous flux is determined 
as the product of the energy flux and the corresponding value of the 
function V (X): 

dO = V(X)dO en (16.37) 

Expressing the energy flux through the function of energy distribu- 
tion by wavelengths [see Eq. (16.35)1, we get 

dO = P (X) <p (X) dX (16.38) 

The total luminous flux is 

oo 

<D= j F(X)<p(X)dX (16.39) 

0 

The function V (X) is a dimensionless quantity. Consequently, the 
dimension of luminous flux coincides with that of energy flux. This 
makes it possible to define the luminous flux as the flux of luminous 
energy assessed according to its visual sensation. 


16.5. Photometric Quantities and Units 

Photometry is the branch of optics occupied in measuring lumi- 
nous fluxes and quantities related to them. 

Luminous Intensity. A source of light whose dimensions may be 
disregarded in comparison with the distance from the place of obser- 
vation to the source is called a point source. In a homogeneous and 
isotropic medium, the wave emitted by a point source will be spher- 
ical. Point sources of light are characterized by the luminous inten- 
sity / determined as the luminous flux emitted by a source per unit 
solid angle: 



General 


331 


(dO is the luminous flux emitted by a source within the limits of the 
solid angle d£2). 

In the general case* the luminous intensity depends on the direc- 
tion: / = / (0, q>) (here 0 and <p are the polar and the azimuth an- 
gles in a spherical system of coordinates). If / does not depend on the 
direction, the light source is called isotropic. For an isotropic source 


/ 


<D 
4 n 


(16.41) 


where O is the total luminous flux emitted by the source in all di- 
rections. 

When dealing with an extended source, we can speak of the lumi- 
nous intensity of an element of its surface dS. Now by d<3> in Eq. 
(16.40) we must understand the luminous flux emitted by the sur- 
face element dS within the limits of the solid angle dQ. 

The unit of luminous intensity — the candela (cd) is one of the basic 
SI units. It is defined as the luminous intensity, in the perpendicular 
direction, of a surface of 1/600 000 square metre of a complete ra- 
diator at the temperature of freezing platinum under a pressure of 
101 325 pascals. By a complete radiator is meant a device having 
the properties of a blackbody (see Vol. III). 

Luminous Flux. The unit of luminous flux is the lumen (lm). It 
equals the luminous flux emitted by an isotropic source with a lu- 
minous intensity of 1 candela within a solid angle of one steradian: 


1 lm = 1 cd'l sr (16.42) 

It has been established experimentally that an energy flux of 
0.0016 W corresponds to a luminous flux of 1 lm formed by radiation 
having a wavelength of X = 0.555 pm. The energy flux 

a >e „=TW W (16.43) 

corresponds to a luminous flux of 1 lm formed by radiation of a 
different wavelength. 

Illuminance. The degree of illumination of a surface by the light 
falling on it is characterized by the quantity 

E= ^nr < 16 - 44 > 

known as the illuminance or illumination (dO lDC is the luminous 
flux incident on the surface element dS ). 

The unit of illuminance is the lux (lx) equal to the illuminance 
produced by a flux of 1 lm uniformly distributed over a surface 
having an area of 1 m a : 

1 lx = 1 lm : 1 m 2 


(16.45) 



332 


Optics 


The illuminance E produced by a point source can be expressed 
through the luminous intensity /, the distance r from the surface 
to the source, and the angle a between a normal to the surface n 
and the direction to the source. The flux incident on the area dS 
(Fig. 16.4) is dd>i nc — I dQ and it is confined within the solid angle dQ 

subtended by d$. The angle dQ is dS cos a/r 2 . 
Hence, dO mc = / dS cos a/r 2 . Dividing this 
flux by d£ f we get 

e== ±c^l (16.46) 

Luminous Emittance. An extended source of 
light can be characterized by the luminous 
emittance M of its various sections, by which 
is meant the luminous flux emitted outward 
by unit area in all directions (within the 
limits of values of 0 from 0 to ji/ 2, where 0 is the angle made by 
the given direction with an external normal to the surface): 

(16.47) 



(d<I>e m is the flux emitted outward in all directions by the surface 
elements dS of the source). 

Luminous emittance may appear as a result of a surface reflecting 
the light falling on it. Here by d( I> em in Eq. (16.47), we must under- 
stand the flux reflected by the surface element dS in all directions. 

The unit of luminous emittance is the lumen per square metre 
(lm/m 2 ). 

Luminance. Luminous emittance characterizes radiation (or 
reflection) of light by a given place of a surface in all directions. The 
radiation (reflection) of light in a given direction is characterized 
by the luminance L . The direction can be given by the polar angle 
0 (measured from the outward normal n to the emitting surface area 
AS) and the azimuth angle (p. Luminance is defined as the ratio of 
the luminous intensity of an elementary surface area AS in a given 
direction to the projection of the area AS onto a plane perpendicu- 
lar to the chosen direction. 

Let us consider the elementary solid angle dQ subtended by the 
luminous area AS and oriented in the direction (0, q>) (Fig. 16.5). 
The luminous intensity of area AS in the given direction, according 
to Eq. (16.40), is / = d<WdQ, where dd> is the luminous flux propa- 
gating within the limits of the angle dQ. The projection of AS onto 
a plane normal to the direction (0, <p) (in Fig. 16.5 the trace of this 
plane is depicted by a dash line) is AS cos 0. Hence, the luminance is 


L 


dd> 

dQAS cos i 


(16.48) 



General 


333 


In the general case, the luminance differs for different directions: 
L = L (0, cp). Like the luminous emittance, the luminance can be 
used to characterize a surface that reflects the light falling on it. 

In accordance with Eq. (16.48), the flux emitted by the area AS 
within the limits of the solid angle dQ in the direction determined 
by 0 and q> is 

dd> = L (0, <p) dQ A S cos 0 (16.49) 

A source whose luminance is identical in all directions (L = const) 
is called a Lambertian source (obeying Lam- 
bert’s law) or a cosine source (the flux emitted 
by a surface element of such a source is pro- 
portional to cos 0). Only a blackbody strictly 
observes Lambert’s law. 

The luminous emittance M and luminance 
L of a Lambertian source are related by a sim- 
ple expression. To find it, let us introduce dQ = 

= sin 0 d0 d 9 into Eq. (16.49) and integrate 
the expression obtained with respect to (p within 
the limits from 0 to 2 n and with respect to 0 
from 0 to jt/2, taking into account that L = 

= const. As a result, we shall find the total light'flux emitted by sur- 
face element A S of a Lambertian source outward in all directions: 



Fig. 16.5 


2n 

A<X> em = L A S j dcp 
0 


ft /2 

j sin 6 cos 0d0 = tlL AS 
o 


We get the luminous emittance by dividing this flux by A S. Thus, 
for a Lambertian source, we have 

M = jiL (16.50) 

The unit of luminance is the candela per square metre (cd/m 2 ). 
A uniformly luminous plane surface has a luminance of 1 cd/m 2 in a 
direction normal to it if in this direction the luminous intensity of 
one square metre of surface is one candela. 


16.6. Geometrical Optics 

The lengths of light waves perceived by the human eye are very 
small (of the order of 10 ~ 7 m). For this reason, the propagation of 
visible light in a first approximation can be considered without giv- 
ing attention to its wave nature and assuming that light propagates 
along lines called rays. In the limiting case corresponding to X — *0, 
the laws of optics can be formulated using the language of geometry. 



334 


Optics 


Accordingly, the branch of optics in which the finiteness of the 
wavelengths is disregarded is known as geometrical optics. Another 
name for it is ray optics. 

Geometrical optics is based on four laws: (1) the law of propagation 
of light along a straight line; (2) the law of independence of light 
rays; (3) the law of light reflection; and (4) the law of refraction. 

The law of straight-line propagation states that in a ho- 
mogeneous medium light propagates in a straight line. This law is ap- 
proximate — when light passes through very 
small openings, deviations from a straight line 
are observed that increase with a diminishing 
size of the opening. 

The law of independence of light rays states 
that rays do not disturb one another when they 
intersect . The intersection of rays does not hin- 
der each of them from propagating indepen- 
dently of the others. This law holds only at 
not too great luminous intensities. At intensi- 
ties reached with the aid of lasers, the inde- 
pendence of light rays stops being observed. 

The laws of reflection and refraction of light were formulated in 
Sec. 16.3 [see Eqs. (16.23) and (16.24) and the text following them]. 

Geometrical optics can be based on the principle established by 
the French mathematician Pierre de Fermat (1601-1665). It under- 
lies the laws of straight-line propagation, reflection, and refraction 
of light. As formulated by Fermat himself, this principle states that 
any light ray will travel between two end points along a line requiring 
the minimum transit time . 

Light needs the time dt — ds/v , where v is the speed of light at 
the given point of the medium, to cover the distance ds (Fig. 16.6). 
Replacing v with c/n [see Eq. (16.2)1, we find that dt — (1/c) n ds. 
Consequently, the time x spent by light in covering the distance 
from point 1 to- point 2 is 

2 

t = -~^ nds (16.51) 

i 

The quantity 

2 

L=Jnds (16.52) 

1 

having the dimension of length is called the optical path. In a homo- 
geneous medium, the optical path equals the product of the geometri- 
cal path s and the index of refraction n of the medium: 

L — ns 


ds ■ 


Fig. 16.6 


(16.53) 



General 


335 


According to Eqs. (16,51) and (16,52), we have 

x-4- (16.54) 

The proportionality of the time x of covering a path to the optical 
path L makes it possible to word Fermat’s principle as follows: 
light travels along a path whose optical length is minimum . More exact- 
ly, the optical path must be extremal, i.e. either minimum or maxi- 
mum, or stationary — identical for all poss- 
ible paths. In the last case, all the paths 
of light between two points are tautochro- 
nous (requiring the same time for covering 
them). 

The reversibility of light rays ensues from 
Fermat’s principle. Indeed, the optical path 
that is minimum when light travels from 
point 1 to point 2 is also minimum when 
light travels in the opposite direction. Con- 
sequently, a ray emitted toward another one 
that has travelled from point 1 to point 2 
will cover the same path, but in the oppo- Fig. 16.7 

site direction. 

Let us use Fermat’s principle to obtain the laws of reflection and 
refraction of light. Assume that a light ray reaches point B from point A 
after being reflected from surface MN (Fig. 16.7, the straight path 
from A to B is blocked by opaque screen Sc). The medium in which 
the ray travels is homogeneous. Therefore, the minimality of the 
optical length consists in the minimality of its geometrical length. 
The geometrical length of an arbitrarily taken path is AO'B = 
= A'O'B (auxiliary point A' is a mirror image of point A). A glance 
at the figure shows that the path of the ray reflected at point O will 
be the shortest. At this point the angle of reflection equals the angle 
of incidence. We must note that when point O' moves away from 
point O y the geometrical path grows unlimitedly so that in the given 
case we have only one extreme — a minimum. 

Now let us find the point at which a ray travelling from A to B 
must be refracted for the optical path to be extremal (Fig. 16.8). 
The optical path for an arbitrary ray is 

L = + n 2 s 2 = n t V a \ + * z + »2 V a\ + (b—x)* 

To find the extreme value, let us differentiate L with respect to x 
and equate the derivative to zero: 



dL _ n Y x n t {b — x) _ n jr n b—x _Q 

dx /«!+** Y * !+(&—*)* 1 *» * ** 



336 


Optics 


The factors of n x and n 2 equal sin 0 and sin 0% respectively. We 
thus get the relation 

n i sin 0 = rc 2 sin 0* 

expressing the law of refraction [see Eq. (16.26)]. 

Let us consider reflection from the inner surface of an ellipsoid of 
revolution (Fig. 16.9; F x and F 2 are the foci of the ellipsoid). Accord- 
ing to the definition of an ellipse, the paths F x OF 2 , F x O'F 2 , F x O”F 2% 
etc. are identical in length. Hence, all the rays leaving focus F x 



Fig. 16.8 



and arriving after reflection at focus F 2 are tautochronous. In this 
case, the optical path is stationary. If we replace the surface of the 
ellipsoid with surface MM having a smaller curvature and oriented 
so that a ray leaving point F x arrives at point F 2 after being reflected 
from MM , then path F x OF % will be minimum. For surface NN whose 
curvature is greater than that of the ellipsoid, path F x OF 2 will be 
maximum. 

The optical paths are also stationary when the rays pass through 
a lens (Fig. 16.10). Ray POP ' has the shortest path in air (where the 
index of refraction n is virtually equal to unity) and the longest path 
in glass ( n « 1.5). Ray PQQ'P' has the longest path in air, but a 
shorter one in glass. As a result, the optical paths will be the same 
for all the rays. Hence, the latter are tautochronous, and the optical 
path is stationary. 

Let us consider a wave propagating in a non-homogeneous isotropic 
medium along rays 7, 2, 5, etc. (Fig. 16.11). We shall consider that 
the non-homogeneity is sufficiently small for us to assume the index 
of refraction to be constant on sections of the rays of length X. We 
shall construct wave surfaces S Xl S 2 , S a , etc. so that the oscillations 
at the points of each following surface lag in phase by 2 n behind 
the oscillations at the points on the preceding surface. The oscilla- 
tions at points on the same ray are described by the equation 5 = 



General 


337 


= A cos (of — kr + a) (here r is the distance measured along the 
ray). The lag in phase is determined by the expression fcAr, where A r 
is the distance between adjacent surfaces. From the condition fcAr = 
= 2n, we find that Ar = 2n/k = X. The optical length of each of 
the paths of geometrical length X is nX = X 0 [see Eq. (16.5)]. Accord- 
ing to Eq. (16.54), the time t during which light covers a path is 




proportional to the optical length of the path. Consequently, the 
equality of the optical paths signifies equality of the times needed 
for light to travel the relevant paths. We thus arrive at the conclu- 
sion that sections of rays confined between two wave surfaces have 
identical optical paths and are tautochronous. In particular, the 
sections of the rays between wave surfaces MM and NN depicted 
by dash lines in Fig. 16.10 are tautochronous. 

It can be seen from our treatment that the lag in phase 6 appearing 
on the optical path L is determined by the expression 

6=-j^2n (16.55) 

(X 0 is the length of a wave in a vacuum). 


16.7. Centered Optical System 

A collection of rays forms a beam. If rays when continued inter- 
sect at one point, the beam is called homocentric. A spherical wave 
surface corresponds to a homocentric beam of rays. Figure 16.12a 
shows a converging, and Fig. 16.126 a diverging homocentric beam. 
A particular case of a homocentric beam is a beam of parallel rays; 
a plane light wave corresponds to it. 


338 


Optics 


Any optical system transforms light beams. If the system does 
not violate the homocentricity of the beams, then the rays emerging 
from point P intersect at one point P\ This point is the optical 
image of point P. If a point of an object is depicted in the form of 
a point, the image is called a point or a stigmatic one. 

An image is called real if the light rays actually intersect at point 
P r (see Fig. 16.12a), and virtual if the continuations of the rays in 

a direction opposite to the direction 
of propagation of the light intersect 
at P ' (see Fig. 16.126). 

Owing to the reversibility of light 
rays, light source P and image P’ 
may exchange roles — a point source 
placed at P' will have its image at 
P. For this reason, P and P' are 
called conjugate points. 

An optical system that produces 
a stigmatic image which.is geomet- 
rically similar to the object it depicts 
is called ideal. With the aid of such a system, a space continuity of 
points P is depicted in the form of a space continuity of points P' m 
The first continuity of points is known as the object space, and the 
second one as the image space. In both spaces, points, straight lines, 
and planes uniquely correspond to one another. Such a relation of 
two spaces is called collinear correspondence in geometry. 

An optical system is a collection of reflecting and refracting sur- 
faces separating optically homogeneous media from one another. 
These surfaces are usually spherical or plane (a plane can be consid- 
ered as a sphere of infinite radius). More complicated surfaces such 
as an ellipsoid, hyperboloid or paraboloid of revolution are used much 
less frequently. 

An optical system formed by spherical (in particular, by plane) 
surfaces is called centered if the centres of all the surfaces are on a 
single straight line. This line is called the optical axis of the system. 

To each point P or plane S in object space there corresponds its 
conjugate point P' or plane S’ in image space. The infinite multitude 
of conjugate points and conjugate planes includes points and planes 
having special properties. Such points and planes are called cardinal 
ones. Among them are the focal, principal, and nodal points and 
planes. Setting of the cardinal points or planes completely deter- 
mines the properties of an ideal centered optical system. 

Focal Planes and Focal Points of an Optical System. Figure 16.13 
shows the external refracting surfaces and the optical axis of an 
ideal centered optical system. Let us take plane S perpendicular 
to the optical axis in the object space of this system. It follows from 
considerations of symmetry that plane S'- conjugate to S is also per- 




General 


339 


pendicular to the optical axis. Displacement of plane S relative to the 
system will produce a corresponding displacement of plane S' . When 
plane S is very far, a further increase in its distance from the system 
will produce virtually no change in the position of plane S'. This 
signifies that as a result of removing plane S to infinity, plane S' 
will be in a definite extreme position F' . Plane F' coinciding with 
the extreme position of plane S' is called the second (or back) focal 
plane of the optical system. We can say briefly that the second focal 
plane F' is defined as a plane conjugate to plane S «*, perpendicular 
to the axis of the system and at infinity in the object space. 

The point of intersection of the second focal plane with the optical 
axis is known as the second (or back) focal point (focus) of the system. 


S F* S' 



Fig . 16.13 


It is also designated by the letter F'. This point is conjugate to point 
Poo on the axis of the system at infinity. Rays emerging from Poo 
form a beam parallel to the axis (see Fig. 16.13). When they leave 
the system, these rays form a beam converging at focal point F\ 
A parallel beam impinging on the system may leave it not in the 
form of a converging beam (as in Fig. 16.13), but in the form of a 
diverging one. Hence, what intersects at point F' will be not the 
actual rays that emerge, but their extensions in the reverse direction. 
Accordingly, the second focal plane will be in front (in the direction 
of the rays) of the system or inside it. 

The rays emanating from an infinitely remote point Q ^ not lying 
on the axis of the system form a parallel beam directed at an angle 
to the axis of the system. Upon emerging from the system, these rays 
form a beam converging at point Q' belonging to the second focal 
plane, but not coinciding with focal point F' (see point Q ' in Fig. 
16.13). It follows from the above that the image of an infinitely remote 
object will be in the focal plane. 

If we remove plane S' perpendicular to the axis to infinity 
(Fig. 16.14), its conjugate plane S will advance to its extreme posi- 
tion F called the first (or front) focal plane of the system. We can say 
for short that the first focal plane F is a plane conjugate to plane S' m 


340 


Optic * 


perpendicular to the axis of the system and at infinity in the image 
space. s 

The point of intersection of first focal plane F with the optical 
axis is called the first (or front) focal point (focus) of the system. 
This point is also designated by the symbol F. The rays emerging 
from focal point F form a beam of rays parallel to the axis after leav- 
ing the system. The rays emerging from point Q belonging to focal 



Fig. 16.14 

plane F (see Fig. 16.14) form a parallel beam directed at an angle 
to the axis of the system after passing through the latter. It may 
happen that a beam which is parallel upon leaving a system is ob- 



Fig. 16.15 

tainecl when a converging beam of light falls on the system instead of 
a diverging one (as in Fig. 16.14). In this case, the first focal point 
is either beyond the system or inside it. 

Principal Planes and Points. Let us consider two conjugate planes 
at right angles to the optical axis of the system. Arrow y (Fig. 16.15) 
in one of these planes will have as its image arrow y ' in the other 
plane. It follows from axial symmetry of the system that arrows y 
and y' must be in the same plane passing through the optical axis 
(in the plane of the drawing). The image y' may be in the same direc- 
tion as object y (see Fig. 16.15a), or in the opposite direction (see 
Fig. 16.156). In the first case, the image is called erect, in the second — 
inverted. Segments laid off upward from an optical axis are considered 
to be positive, and those laid off downward— -negative. The actual 
lengths of the segments are shown in drawings, i.e. the positive 
quantities (— y) and (—y‘) for negative segments. 


General 


341 


The ratio of the linear dimensions of an image and an object is 
called the linear (longitudinal) or the lateral magnification. Designat- 
ing it by the symbol M , we can write 

= (16.56) 

The linear magnification is an algebraic quantity. It is positive if 
the image is erect (the signs of y and y' are the same) and negative 
if the image is inverted (the signs of y 
and y ' are opposite). 

We can prove that there are two con- 
jugate planes which reflect each other 
with a linear magnification of M = + l. 

These planes are known as the princi- 
pal ones. The plane belonging to the 
object space is called the first (or front) 
principal plane of a system. It is des- 
ignated by the symbol H. The plane 
belonging to the image space is called 
the second (or back) principal plane. 

Its symbol is //'. The points of inter- 
section of the principal planes with 
the optical axis are called the prin- 
cipal points of the system (first and 
second, respectively). They are desig- 
nated by the same symbols H and H'. 

Depending on the design of a system, 
its principal planes and points may 
be either outside or inside the system. 

One of the planes may be outside and the other inside a system. 
Finally, both planes may be outside a system at the same side of it. 

It can be seen from the definition of the principal planes that ray 1 
intersecting (actually — Fig. 16.16a, or when virtually continued inside 
the system — Fig. 16.166) the first principal plane H at point Q has 
as its conjugate ray V that intersects (directly or upon virtual con- 
tinuation) principal plane H' at point Q\ The latter is in the same 
direction and at the same distance from the axis as point Q . This is 
easy to understand if we remember that Q and Q' are conjugate points, 
and take into account that any ray passing through point Q must 
have as its conjugate a ray passing through point Q'. 

Nodal Planes and Nodal Points. Conjugate points N and N' lying 
on the optical axis and having the property that the conjugate rays 
passing through them (actually or when imaginarily continued inside 
the system) are parallel to each other are called nodal points or nodes 
(see rays 1~1’ and 2-2' in Fig. 16.17). Planes perpendicular to the 


H H ’ 

I fj W l„ 


a_ 

L U...X 


a' 

_ 


f 1 




1 1 




(*) 



(b) 

Fig. 16.16 



342 


OptiCM 


axis and passing through the nodal points are called nodal planes 
(first and second). 

The distance between the nodal points always equals that between 
the principal points. When the optical properties of the media at 

both sides of the system are the same 
(i.e. n = /i'), the nodal and principal points 
coincide. 

Focal Lengths and Optical Power of a Sys- 
tem. The distance from first principal point 
H to first focal point F is called the first 
focal length f of the system. The distance 
from //' to F' is known as the second focal 
length /'. The focal lengths / and /', are 
algebraic quantities. They are positive if a 
given focal point is at the right of the rel- 
evant principal point, and negative in the opposite case. For ex- 
ample, for the system shown in Fig. 16.18 (see below), the second 
focal length /' is positive, and the first focal length / is negative. 




The figure depicts the true length of HF y i.e. the positive quantity 
( — /) equal to the absolute value of /. 

We can show that the following relation holds between the focal 
lengths / and /' of a centered optical system formed by spherical 
refracting surfaces: 



(16.57) 


where n is the refractive index of the medium in front of the optical 
system, and n' is the refractive index of the medium behind the sys- 
tem. Examination of Eq. (16.57) shows that when the refractive indi- 
ces of the media at both sides of an optical system are the same, the 
focal lengths differ only in their sign: 

r =-/ 


(16.58) 



General 


343 


The quantity 

P = -£-=— -j- (16.59) 


is known as the optical power of a system. When P grows, the focal 
length /' diminishes, and, consequently, the rays are refracted by 
the optical system to a greater extent. The optical power is measured 
in dioptres (D). To obtain P in dioptres, the focal length in Eq. (16.59) 
must be taken in metres. When P is positive, the second focal length /' 
is also positive; hence, the system produces a real image of an infi- 
nitely remote point — a parallel beam of rays is transformed into a 
converging one. In this case, the system is called converging. When P 
is negative, the image of an infinitely remote point will be virtual — a 
parallel beam of rays is transformed by the system into a diverging 
one. Such a system is called diverging. 

Formula of a System. We completely determine the properties of 
an optical system by setting its cardinal planes or poin ts. In parti- 
cular, knowing the position of the cardinal planes, we can construct 
the optical image produced by a system. Let us tak e segment OP 
perpendicular to the optical axis in the object space (Fig. 16.18, 
the nodal points are not shown in the figure). The position of this 
segment can be set either by the distance x measured from point F 
to point O , or by the distance s from H to O. The quantities x and s, 
like the focal lengths / and /', are algebraic ones (their magnitudes 
are shown in figures). 

Let us draw ray 1 parallel to the optical axis from point P. It will 
intersect plane H at point A . In accordance with the properties of 
principal planes, ray V conjugate to ray 1 must pass through point A f 
of plane H' conjugate to point A. Since ray 1 is parallel to the optical 
axis, then ray 2' conjugate to it will pass through second focal point F'. 
Now let us draw ray 2 passing through the first focal point F from 
point P. It will intersect plane H at point B . Ray 2 9 conjugate to 
it will pass through point B ' of plane H f conjugate to B and will be 
parallel to the optical axis. Point P' of intersection of rays V and 2 9 
is the image of point P. Image O'P', like object OP, is perpendicular 
to the optical axis. 

The position of image O'P 9 can be characterized either by the 
distance x* from point F 9 to point O 9 or by the distance s’ from H' 
to O'. The quantities x' and s' are algebraic ones. For the case shown 
in Fig. 16.18, they are positive. 

The quantity x r determining the position of the image is related 
to the quantity x determining the position of the object and to the 
focal lengths / and /'. For the right triangles with a common apex 
at point F (see Fig. 16.18), we can write the relation 


OP _ 
HB ~ 



( 16 . 60 ) 



344 


Optics 


Similarly, for the triangles with their common apex at point F\ 
we have 


R' A' y /' 
0'P' ~ —y' ~~ x' 


(16.61) 


Combining both relations, we find that (— x)/( — f) = f/x\ whence 


= //' (16.62) 

This equation is known as Newton’s formula. For the condition that 
n — n\ Newton’s formula has the form 


xx 7 = — /* (16.63) 

[see Eq. (16.57)]. 

It is easy to pass over from the formula relating the distances x 
and x 7 to the object and to the image from the focal points of a system 
to a formula establishing the relation between the distances s and s 7 
from the principal points. A glance at Fig. 16.18 shows that ( — x) = 
= ( — s) — ( — /) (i.e. x = s — /), and x’ = s' — /'. Introducing 
these expressions for x and x 7 into Eq. (16.62) and making the rele- 
vant transformations, we get 

-f + T- =1 (*6.64) 


When the condition is observed that f = — / [see Eq. (16.58)], 
Eq. (16.64) is simplified as follows: 


s 


s' 



(16.65) 


Equations (16.62)-(16.65) are equations of a centered optical system. 


16.8. Thin Lens 


A lens is a very simple centered optical system. It is a transparent 
(usually glass) body bounded by two spherical surfaces* (in a parti- 
cular case one of the surfaces can be plane). The points of intersec- 
tion of the surfaces with the optical axis of a lens are called the apices 
of the refracting surfaces. The distance between the apices is named 
the thickness of the lens. If the lens thickness may be ignored in com- 
parison with the smaller of the radii of curvature of the surfaces bound- 
ing a lens, the latter is called thin. 

Calculations which we do not give here show that for a thin lens 
the principal planes H and H' may be considered to coincide and 
pass through the centre O of the lens (Fig. 16.19). The following ex- 
pression is obtained for the focal lengths of a thin lens: 


/'=- 


n — n 0 R t —R t 


(16.66) 


* There are also lenses with surfaces having a more intricate shape. 



General 


345 


where n = refractive index of the lens 

n 0 = refractive index of the medium surrounding the lens 
R t and i? 2 = radii of curvature of the lens surfaces. 

The radii of curvature must be treated as algebraic quantities: for 
a convex surface (i.e. when the centre of curvature is to the right of 
the apex), the radius of curvature 
must be considered positive, and for 
a concave surface (i.e. when the cen- 
tre of curvature is to the left of the 
apex) the radius must be considered 
negative. The magnitude of the ra- 
dius of curvature is shown in draw- 
ings, i.e. — R if R <C 0. 

If the refractive indices of the 
media at both sides of a thin lens 
are the same, then the nodal points 
N and N' coincide with the prin- 
cipal points, i.e. are at the centre O of the lens. Hence, in this case, 
any ray passing through the centre of the lens does not change its 
direction. If the refractive indices of the media before and after a 

lens are different, then the nodal points 
do not coincide with the principal 
points and a ray passing through the 
centre of the lens changes its direction. 

A parallel beam of rays after passing 
through a lens converges at a point on 
the focal plane (see point Q r in Fig. 
16.20). To determine the position of 
this point, we must continue the ray 
passing through the centre of the lens 
up to its intersection with the focal 
plane (see ray OQ' shown by a dash line). 
The other rays will gather at the point 
of intersection too. Such a method is 
suitable when the optical properties of 
the medium at each side of a lens are 
identical ( n —n t ). Otherwise a ray pass- 
ing through the centre will change its direction. To find point Q' in 
this case, we must know the position of the nodal points of the lens. 

We must note that the optical paths laid off along the rays, begin- 
ning at wave surface SS (see Fig. 16.20) and terminating at point Q' 
are identical and are tautochronous (see the end of Sec. 16.6). 

In concluding, we must say that a lens is a far from ideal optical 
system. The images of objects it produces have a number of errors. 
But a consideration of them is beyond the scope of the present 
book. 




Fig. 16.19 


346 


Optics 


16.9. Huygens’ Principle 

In the following two chapters, we shall have to do with processes 
taking place behind an opaque barrier with apertures when a light 
wave impinges on the barrier. In the approximation of geometrical 
optics, no light ought to penetrate beyond the barrier into the region 
of the geometrical shadow. Actually, however, a light wave in prin- 
ciple propagates throughout the entire space behind the barrier and 
penetrates into the region of the geometrical shadow, this penetra- 
tion being the more noticeable, the smaller are the dimensions of 





!♦ 


+ ♦ 


Fig. 16.22 


the apertures. With a diameter of the apertures or a width of slits 
comparable with the length of a light wave, the approximation of 
geometrical optics is absolutely illegitimate. 

The behaviour of light behind a barrier with an aperture can be 
explained qualitatively with the aid of Huygens’ principle, named 
in honour of the Dutch physicist Christian Huygens (1629-1696) who 
discovered it. This principle establishes the way of constructing 
a wavefront at the moment of time t + At according to the known 
position of the wavefront at the moment t . According to Huygens’ 
principle, every point on an advancing wavefront can be considered 
as a source of secondary wavelets, and the envelope of these wavelets 
defines a new wavefront (Fig. 16.21; the medium is assumed to be 
non-homogeneous — the velocity of the wave in the lower part of the 
figure is greater than in the upper one). 

Assume that a plane barrier with an aperture is struck by a wave- 
front parallel to it (Fig. 16.22). According to Huygens, every point 
on the portion of the wavefront bordering on the aperture is a centre 
of secondary wavelets which will be spherical in a homogeneous and 
isotropic medium. Constructing the envelope of these wavelets, we 
shall see that the wave penetrates beyond the aperture into the region 


General 


347 


of the geometrical shadow (these regions are shown by dash lines in the 
figure), bending around the edges of the barrier. 

Huygens’ principle gives no information on the intensity of waves 
propagating in various directions. This shortcoming was eliminated 
by the French physicist Augustin Fresnel (1788-1827). The improved 
Huygens-Fresnel principle is treated in Sec. 18.1, where a physical 
substantiation of the principle is also given. 



CHAPTER 17 INTERFERENCE OF LIGHT 


17.1. Interference of Light Waves 

Let us assume that two waves of the same frequency, being super- 
posed on each other, produce oscillations of the same direction, name- 
ly, 

A Y cos (o t + ay); A 2 cos (<ot + a 2 ) 

at a certain point in space. The amplitude of the resultant oscilla- 
tion at the given point is determined by the expression 

A 2 = A\ + A\ + 2A 1 A 2 T cos 6 

where 6 = a 2 — ay [see Eq. (7.84) of Vol. I, p. 2041. 

If the phase difference 6 of the oscillations set up by the waves 
remains constant in time, then the waves are called coherent*. 

The phase difference 6 for incoherent waves varies continuously 
and takes on any values with an equal probability. 1 Hence, the time- 
averaged value of cos 6 equals zero. Therefore 

(A*) = (A*) + (Al) 

Taking into account Eq. (16.10), we thus conclude that the intensity 
observed upon the superposition of incoherent waves equals the sum 
of the intensities produced by each of the waves individually: 

/=/, + /* (17.1) 

For coherent waves, cos 6 has a time-constant value (but a differ* 
ent one for each point of space), so that 

J = /, + / a + 2l/77^cos8 (17.2) 

At the points of space for which cos 6 > 0, the intensity / will ex- 
ceed Iy + / 2 ; at the points for which cos£ < 0, it will be smaller 
than I x + / 2 . Thus, the superposition of coherent light waves is 
attended by redistribution of the light flux in space. As a result, 
maxima of the intensity will appear at some spots and minima at 
others. This phenomenon is called the interference of waves. Inter- 
ference manifests itself especially clearly when the intensity of both 
interfering waves is the same: I x = / 2 . Hence, according to Eq. (17.2), 


* We shall discuss the concept of coherence in greater detail in the following 



Interference of Light 


349 


at the maxima I = 4/ lt while at the minima 1=0. For incoherent 
waves in the same condition, we get the same intensity I = 2 I t 
everywhere [see Eq. (17.1)]. 

It follows from what has been said above that when a surface is 
illuminated by several sources of light (for example, by two lamps), 
an interference pattern ought to be observed with a characteristic 
alternation of maxima and minima of intensity. We know from our 
everyday experience, however, that in this case the illumination of 




the surface diminishes monotonously with an increasing distance from 
the light sources, and no interference pattern is observed. The expla- 
nation is that natural light sources are not coherent. 

The incoherence of natural light sources is due to the fact that the 
radiation of a luminous body consists of the waves emitted by many 
atoms. The individual atoms emit wave trains with a duration of 
about 10“® s and a length of about 3 m (see Sec. 16.1). The phase of 
a new train is not related in any way to that of the preceding one. 
In the light wave emitted by a body, the radiation of one group of 
atoms after about lO^ 8 s is replaced by the radiation of another group, 
and the phase of the resultant wave undergoes random changes. 

Coherent light waves can be obtained by splitting (by means of 
reflections or refractions) the wave emitted by a single source into 
two parts. If these waves are made to cover different optical paths 
and are then superposed onto each other, interference is observed. 
The difference between the optical paths covered by the interfering 
waves must not be very great because the oscillations being added 
must belong to the same resultant wave train. If this difference will 
be of the order of one metre, oscillations corresponding to different 
trains will be superposed, and the phase difference between them will 
continuously change in a chaotic way. 

Assume that the splitting into two coherent waves occurs at point O 
(Fig. 17.1). Up to point P, the first wave travels the path s l in a me- 


350 


Optics 


dium of refractive index n l9 and the second wave travels the path s t 
in a medium of refractive index n t . If the phase of oscillations at 
point O is < of, then the first wave will produce the oscillation 
A x cos (o (t — point P, and the second wave, the oscillation 

A 2 cos (D (t — sjv 2 ) at this point; = c/n x and v % = c/n 2 are the 
phase velocities of the waves. Hence, the difference between the 
phases of the oscillations produced by the waves at point P will be 

Replacing c o/c with 2nv/c =2 n/X 0 (where is the wavelength in 
a vacuum), the expression for the phase difference can be written in 
the form 

(17.3) 

where 

A = n^s % — *i*i = L t — Lx (17.4) 

is a quantity equal to the difference between the optical paths trav- 
elled by the waves and is called the difference in optical path [com- 
pare with Eq. (16.55)]. 

A glance at Eq. (17.3) shows that if the difference in the optical 
path equals an integral number of wavelengths in a vacuum: 

A = ±mX 0 (m = 0, 1, 2, . . .) (17.5) 

then the phase difference 6 is a multiple of 2 ji, and the oscillations 
produced at point P by both waves will occur with the same phase. 
Thus, Eq. (17.5) is the condition for an interference maximum, i.e. 
for constructive interference. 

If A equals a half-integral number of wavelengths in a vacuum: 

A=±(w + -l)x 0 (m = 0, 1, 2,...) (17.6) 

then 6 = ±(2 m + 1) n, so that the oscillations at point P are in 
counterphase. Thus, Eq. (17.6) is the condition for an interference 
minimum, i.e. for destructive interference. 

Let us consider two cylindrical coherent light waves emerging 
from sources and S 2 having the form of parallel thin luminous 
filaments or narrow slits (Fig. 17.2). The region in which these waves 
overlap is called the interference field. Within this entire region, there 
are observed alternating places with maximum and minimum inten- 
sity of light. If we introduce a screen into the interference field, we 
shall see on it an interference pattern having the form of alternating 
light and dark fringes. Let us calculate the width of these ^fringes, 
assuming that the screen is parallel to a plane passing through sources 
Si and S % . We shall characterize the position of a point on the screen 



Interference of Light 


351 


by the coordinate x measured in a direction at right angles to lines S t 
and S 2 . We shall choose the beginning of our readings at point O 
relative to which S x and S 2 are arranged symmetrically. We shall 
consider that the sources oscillate in the same phase. Examination 
of Fig. 17.2 shows that 

*‘ = ZM-(*-4) 2 ; s; = * 2 +(* + 4) 2 

Hence, 

s\ — s » = (s 2 + Si) (s 2 — Si) = 2 xd 

It will be established somewhat later that to obtain a distinguish- 
able interference pattern, the distance between the sources d must 
be considerably smaller than the distance to the screen 1. The dis- 
tance x within whose limits interference fringes are formed is also 
considerably smaller than L In these conditions, we can assume that 
s 2 + s t « 21. Thus, s 2 — s x = xd/l. Multiplying s 2 — s x by the 
refractive index of the medium n, we get the difference in the optical 
path 

A = n~ (17.7) 

The introduction of this value of A into condition (17.5) shows that 
intensity maxima will be observed at values of x equal to 

*max = ± in ~ X (m = 0,1,2,...) (17.8) 

Here X = X 0 /n is the wavelength in the medium filling the space 
between the sources and the screen. 

Using the value of A given by Eq. (17.7) in condition (17.6), we 
get the coordinates of the intensity minima: 

2 m in= ±(m+4) ( m = °. 1, 2, . . .) (17.9) 

Let us call the distance between two adjacent intensity maxima 
the distance between interference fringes, and the distance between 
adjacent intensity minima the width of an interference fringe. It can 
be seen from Eqs. (17.8) and (17.9) that the distance between fringes 
and the width of a fringe have the same value equal to 

Ax = -^X (17.10) 

According to Eq. (17.10), the distance between the fringes grows 
with a decreasing distance d between the sources. If d were compa- 
rable with Z, the distance between the fringes would be of the same 
order as X, i.e. would be several scores of micrometres. In this case, 
the separate fringes would be absolutely indistinguishable. For an 


352 


Optta 


interference pattern to become distinct, the above-mentioned con- 
dition d l must be observed. 

If the intensity of the interfering waves is the same (I x = 1 2 =- / 0 ) ? 
then according to Eq. (17.2) the resultant intensity at the points 
for which the phase difference is 6 is determined by the expression 

I = 21 o (1 + cos 6) 4/ 0 cos 2 -y 

Since 8 is proportional to A (see Eq. (17.3)1, then in accordance with 
Eq. (17.7) 6 grows proportionally to x. Hence, the intensity varies 

along the screen in accordance with the law 
of cosine square. The right-hand part of 
Fig. 17.2 shows the dependence of / on x ob- 
tained in monochromatic light. 

The width of the interference fringes and 
their spacing depend on the wavelength X. 
The maxima of all wavelengths will coincide 
only at the centre of a pattern when x = 0. 
With an increasing distance from the centre 
of the pattern, the maxima of different col- 
ours become displaced from one another 
more and more. The result is blurring of 
the interference pattern when it is observed in white light. The 
number of distinguishable interference fringes appreciably grows 
in monochromatic light. 

Having measured the distance between the fringes Ax and know- 
ing l and d, we can use Eq. (17.10) to find X. It is exactly from exper- 
iments involving the interference of light that the wavelengths for 
light rays of various colours were determined for the first time. 

We have considered the interference of two cylindrical waves. 
Let us see what happens when two plane waves are superposed. As- 
sume that the amplitudes of these waves are the same, and the 
directions of their propagation make the angle 2<p (Fig. "47.3). We 
shall consider that the directions of oscillations of the light vector 
are perpendicular to the plane of the drawing. The wave vectors k ( 
and k 2 are in the plane of the drawing and have the same magnitude 
equal to k = 2n/X. Let us write the equations of these waves: 

A cos (c ot — k 4 r) = A cos (a U — A: sin <p x — A: cos <p -y) 

A cos (cot — k 2 r) = A cos (cot -f- k sin <p • x — k cos qp • y) 

The resultant oscillation at points with the coordinates x and y 
has the form 

A cos ( ( ot — k si n qp • x — k cos <p • y ) + A cos (a)*-(-&sinqp-x — k cos <p • y) = 

2 A cos (A: sin <p-x) cos (&t — k cos<p-y) (17.11) 




Interference of Light 


353 


It follows from this equation that at points where k sin <p-x = 
= dz mn (; m =0, 1, 2, . . .), the amplitude of the oscillations 

is 2 A ; where k sin <p*x = + the amplitude of the oscil- 

lations is zero. No matter where we place screen Sc, which is per- 
pendicular to the y-axis, we shall observe on it a system of alternat- 
ing light and dark fringes parallel to the z-axis (this axis is perpen- 
dicular to the plane of the drawing). The coordinates of the intensity 
maxima will be 




= ± 


mn 

k sin <p 


= ± 


mX 

2 sin <p 


(17.12) 


Only the phase of the oscillations depends on the position of the 
screen (on the coordinate y) [see Eq. (17.11)]. 

We have assumed for simplicity that the initial phases of inter- 
fering waves are zero. If the difference between these phases is other 
than zero, a constant addend will appear in Eq. (17.12) — the fringe 
pattern will move along the screen. 


17.2. Coherence 

By coherence is meant the coordinated proceeding of several oscil- 
latory or wave processes. The degree of coordination may vary. 
We can accordingly introduce the concept of the degree of coherence 
of two waves. 

Temporal and spatial coherence are distinguished. We shall begin 
with a discussion of temporal coherence. 

Temporal Coherence. The process of interference described in the 
preceding section is idealized. This process is actually much more 
complicated. The reason is that a monochromatic wave described 
by the expression 

A cos (cot — kr + a) 

where A , co, and a are constants, is an abstraction. A real light wave 
is formed by the superposition of oscillations of all possible frequen- 
cies (or wavelengths) confined within a more or less narrow but finite 
range of frequencies Aco (or the corresponding range of wavelengths 
AX). Even for light considered to be monochromatic (single-coloured), 
the frequency interval Aco is finite*. In addition, the amplitude of 
the wave A and the phase a undergo continuous random (chaotic) 
changes with time. Hence, the oscillations produced at a certain point 
of space by two superposed light waves have the form 

A x ( t ) cos [o)j (t) t + ^ (*)]; A 2 (t) cos [co 2 (*)•* + a 2 (t)] (17.13) 

• The spectral lines emitted by atoms have a “natural” width of the order of 
10* rad/s (AX~ 10“ 4 A). 


354 


Optics 


the chaotic changes in the functions A x (*), & x (*), a x (*), A r (t ), 
co 2 ( t ), and a 2 (<) being absolutely independent. 

We shall assume for simplicity’s sake that the amplitudes A x 
and A 2 are constant. Changes in the frequency and phase can be re- 
duced either to a change only in the phase, or to a change only in the 
frequency. Let us write the function 

/ (0 = A cos [a) (*)•< + a (t) ) (17.14) 

in the form 

f (t) = A cos {c o 0 t + [co (*) — co 0 l t + a (f)} 

where co 0 is a certain average value of the frequency, and introduce 
the notation [co (<) — co 0 l t + a (t) = a' ( t ). Equation (17.14) will 
thus become 

f(t) = A cos [co 0 ^+-a'-(i)] (17.15) 

We have obtained a function in which only the phase of the oscilla- 
tion changes chaotically. 

On the other hand, it is proved in mathematics that an inharmonic 
function, for example, function (17.14), can be represented in the 
form of the sum of harmonic functions with frequencies confined within 
a certain interval Aco [see Eq. (17.16)). 

Thus, when considering the matter of coherence, two approaches 
are possible: a “phase” one and a “frequency” one. Let us begin with 
the phase approach. Assume that the frequencies & x and co 2 in Eqs. 
(17.13) satisfy the condition (Dj = co 2 = const. Now let us find the 
influence of a change in the phases a x and a 2 . According to Eq. (17.2), 
with our assumptions, the intensity of light at a given point is deter- 
mined by the expression 

I = I i + I z + 2V7TzCos6(t) 

where 6 (t) — a 2 (t) — <Xj (*). The last addend in this equation is 
called the interference term. 

An instrument that can be used to observe an interference pattern 
(the eye 3 *, a photographic plate, etc.) has a certain inertia. In this 
connection, it registers a pattern averaged over the time interval 
finstr needed for “operation” of the instrument. If during the time 
*instr the factor cos 6 (t) takes on all the values from — 1 to +1, the 
average value of the interference term will be zero. Therefore, the 
intensity registered by the instrument will equal the sum of the 
intensities produced at a given point by each of the waves separately — 
interference is absent, and we are forced to acknowledge that the 
waves are incoherent. 

* We remind our reader that the showing of motion picture films is based on 
the inertia of visual perception, which is about 0.1 second. 


Interference of Light 


355 


If during the time *in St r> however, the value of cos fi (t) remains 
virtually constant*, the instrument will detect interference, and 
the waves must be acknowledged as coherent. 

It follows from the above that the concept of coherence is relative: 
two waves can behave like coherent ones when observed using one 
instrument (having a low inertia), and like incoherent ones when 
observed using another instrument (having a high inertia). The coher- 
ent properties of waves are characterized by introducing the coher- 
ence time tcoh • If is defined as the time during which a chance 
change in the wave phase a (*) reaches a value of the order of n. 
During the time an oscillation, as it were, forgets its initial 

phase and becomes incoherent with respect to itself. 

Using the concept of the coherence time, we can say that when the 
instrument time is much greater than the coherence time of the super- 
posed waves (£mstr £coh)» the instrument does not register inter- 
ference. When t iQ S tr t cohy the instrument will detect a sharp 
interference pattern. At intermediate values of t m s tr, the sharpness 
of the pattern will diminish as t mstr grows from values smaller than 
f coh to values greater than it. 

The distance Z CO h = c2 coh over which a wave travels during the time 
*coh is called the coherence length (or the train length). The coherence 
length is the distance over which a chance change in the phase reaches 
a value of about n. To obtain an interference pattern by splitting 
a natural wave into two parts, it is essential that the optical path 
difference A be smaller than the coherence length. This requirement 
limits the number of visible interference fringes observed when using 
the layout shown in Fig. 17.2. An increase in the fringe number m 
is attended by a growth in the path difference. As a result, the sharp- 
ness of the fringes becomes poorer and poorer. 

Let us pass over to a consideration of the part of the non-monochro- 
matic nature of light waves. Assume that light consists of a sequence 
of identical trains of frequency co 0 and duration t. When one train is 
replaced with another one, the phase experiences disordered changes. 
As a result, the trains are mutually incoherent. With these assump- 
tions, the duration of a train t virtually coincides with the coherence 
ti me fcoh* 

In mathematics, the Fourier theorem is proved, according to which 
any finite and integrable function F (t) can be represented in the 
form of the sum of an infinite number of harmonic components with 
a continuously changing frequency: 

+ oo 

F(t)= | A (<d) (17.16) 


• The phase difference 6 (f) varies for different points of space. The influence 
of the interference term manifests itself at the points where it differs from zero. 


356 


Optics 


Expression (17.16) is known as the Fourier integral. The function 
A (m) inside the integral is the amplitude of the relevant monochro- 
matic component. According to the theory of Fourier integrals, the 
analytical form of the function A (co) is determined by the expression 

4-00 

A(a) = 2n j F (!)«-*"*<$ (17.17) 

— OD 

where £ is an auxiliary integration variable. 

Assume that the function F ( t ) describes a light disturbance at a 
certain point at the moment of time t due to a single wave train. 
Hence, it is determined by the conditions 

F (t) = A 0 exp (iconO at 1 1 1 

F(t) = 0 at Ji| > -J- 

A graph of the real part of this function is given in Fig. 17.4. 


m. j 

A A A A A 

\ A 

R] 

2X 

A At ^ 


V V V V V 

o\J X 

r 


\J \j * 

-t/2 

+vj2 


Fig. 17.4 


Outside the interval from — x/2 to +t/ 2, the function F ( t ) is 
zero. Therefore, expression (17.17) determining the amplitude of the 
harmonic components has the form 

4-T/2 

A (co) — 2 ji j [A 0 exp (ico 0 £)] exp ( — ico£) dl — 

-t/2 

4- t/2 

= 2nA 0 J [exp i (o) 0 - e>) \\ dl = 2n A 0 

-t/2 


After introducing the integration limits and simple transformations, 
we arrive at the equation 


A (co) = nA 0 x 


sin [(o> — fa> 0 ) t/ 2] 
((co — do) t/2] 



Interference of Light 


357 


The intensity I (<o) of a harmonic wave component is proportional 
to the square of the amplitude, i.e. to the expression 


/(«>)= 


sin* [(( 0 — cdq)t/ 2] 
l(co — co 0 ) x/2] a 


(17.18) 


A graph of function (17.18) is shown in Fig. 17.5. A glance at the 
figure shows that the intensity of the components whose frequencies 



are within the interval of width Aco == 2n/x considerably exceeds 
the intensity of the remaining components. This circumstance allows 
us to relate the duration of a train x to the effective frequency range 
A© of a Fourier spectrum: 



Identifying x with the coherence time, we arrive at the relation 

fcoh - (17.19) 

(The sign ~ stands for “equal to in the order of magnitude”). 

It can be seen from expression (17.19) that the broader the interval 
of frequencies present in a given light wave, the smaller is the co- 
herence time of this wave. 

The frequency is related to the wavelength in a vacuum by the 
expression v = c/X 0 . Differentiation of this expression yields Av = 
= cAX<>/Xg « cAX/X 2 (we have omitted the minus sign obtained in 
differentiation and also assumed that X 0 « X). Substituting for Av 
in Eq. (17.19) its expression through X and AX, we obtain the fol- 
lowing expression for the coherence time: 

* coh ~ Tax 


(17.20) 


358 


Optics 


Hence, we get the following value for the coherence length: 

^coh =: ^coli^“JJ' (17.21) 

Examination of Eq. (17.5) shows that the path difference at which 
a maximum of the m-th order is obtained is determined by the relation 

A m — i/M-Xp ~f~ fftX 

When this path difference reaches values of the order of the coherence 



Fig. 17.6 

length, the fringes become indistinguishable. Consequently, the 
extreme interference order observed is determined by the condition 

TOextA ~ Z cob ~ 

whence 

mextr ~TT (17.22) 

It follows from Eq. (17.22) that the number of interference fringes 
observed according to the layout shown in Fig. 17.2 grows when 
the wavelength interval in the light used diminishes. y 

Spatial Coherence. According to the equation k = co/i; = nco/c,. 
scattering of the frequencies Ao results in scattering of the values of 
k. We have established that the temporal coherence is determined 
by the value of Aco. Consequently, the temporal coherence is asso- 
ciated with scattering of the values of the magnitude of the wave 
vector k. Spatial coherence is associated with scattering of the direc- 
tions of the vector k that is characterized by the quantity Ae*. 

The setting up at a certain point of space of oscillations produced 
by waves with different values of e* is possible if these waves are 
emitted by different sections of an extended (not a point) light source. 
Let us assume for simplicity’s sake that the source has the form of 
a disk visible from a given point at the angle <p. It can be seen from 
Fig. 17.6 that the angle <p characterizes the interval confining the 
unit vectors e*. We shall consider that this angle is small. 

Assume that the light from the source falls on two narrow slits 
behind which there is a screen (Fig. 17.7). We shall consider that 
the interval of frequencies emitted by the source is very small. This 
is needed for the degree of temporal coherence to be sufficient for 



Interference of Light 


359 


obtaining a sharp interference pattern. The wave arriving from the 
section of the surface designated in Fig. 17.7 by O produces a zero- 
order maximum M at the middle of the screen. The zero-order maxi- 
mum M ' produced by the wave arriving from section O' will be dis- 
placed from the middle of the screen by the distance x'. Owing to the 
smallness of the angle <p and of the ratio <///, we can consider that 



x ' = /(p/2. The zero-order maximum M ” produced by the wave arriv- 
ing from section O m is displaced in the opposite direction from the 
middle of the screen over the distance x” equal to x'. The zero-order 
maxima from the other sections of the source will be between the 
maxima M ' and M*. 

The separate sections of the light source produce waves whose phases 
are in no way, related to one another. For this reason, the interference 
pattern appearing on the screen will be a superposition of the pat- 
terns produced by each section separately. If the displacement x' 
is much smaller than the width of an interference fringe Ax = Wd 
Isee Eq. (17.10)], then the maxima from different sections of the 
source will practically be superposed on one another, and the pattern 
will be like one produced by a point source. Whin x r Ax, the 
maxima from some sections will coincide with the minima from others, 
and no interference pattern will be observed. Thus, an interference 
pattern will be distinguishable provided that x' < Ax, i.e. 

(17.23) 

or 

q><^ (17-24) 

We have omitted the factor 2 when passing over from expression 
(17.23) to (17.24). 


360 


Optics 


Formula (17.24) determines the angular dimensions of a source 
at which interference is observed. We can also use this formula to 
find the greatest distance between the slits at which interference 
from a source with the angular dimension <p can still be observed. 
Multiplying inequality (17.24) by d/tp, we arrive at the condition 

d<A (17.25) 

A collection of waves with different values of e h can be replaced 
with the resultant wave falling on a screen with slits. The absence 
of an interference pattern signifies that the oscillations produced by 
this wave at the places where the first and second slits are situated 
are incoherent. Consequently, the oscillations in the wave itself at 
points at a distance d apart are incoherent too. If the source were 
ideally monochromatic (this means that Av = 0 and f coh = oo), 
the surface passing through the slits would be a wave one, and 
the oscillations at all the points of this surface would occur in the 
same phase. We have established that when Av = 5 ^ 0 and the source 
has finite dimensions (<p ¥= 0 ), the oscillations at points of a surface 
at a distance of d >X/<p are incoherent. 

We shall call a surface which would be a wave one if the source 
were monochromatic a pseudowave surface* for brevity. We could 
satisfy condition (17.24) by reducing the distance d between the 
slits, i.e. by Taking closer points of the pseudowave surface. Conse- 
quently, oscillations produced by a wave at adequately close points 
of a pseudowave surface are coherent. Such coherence is called spatial. 

Thus, the phase of an oscillation changes chaotically when passing 
from one point of a pseudowave surface to another. Let us introduce 
the distance p CO hi upon displacement by which along a pseudowave 
surface a random change in the phase reaches a value of about n. 
Oscillations at two points of a pseudowave surface spaced apart at 
a distance less than will be approximately coherent. The dis- 
tance pcoh is called the spatial coherence length or the coherence radius. 
It can be seen from expression (17.25) that 

Pcoh~-£ (17.26) 

The angular dimension of the Sun is about 0.01 radian, and the 
length of its light waves is about 0.5 pm. Hence, the coherence radius 
of the light waves arriving from the Sun has a value of the order of 

p c °h = = 50 pm = 0.05 mm (17.27) 


* It must be borne in mind that this term is not used in scientific publica- 
tions. The author has coined it for conditional use only to make the treatment 
more illustrative. 



Interference of Light 


361 


The entire space occupied by a wave can be divided into parts in 
each of which the wave approximately retains coherence. The volume 
of such a part of space, called the coherence volume, in its order of 
magnitude equals the product of the temporal coherence length 
and the area of a circle of radius p C oh- 

The spatial coherence of a light wave near the surface of the heated 
body emitting it is restricted by a value of p CO h of only a few wave- 
lengths. With an increasing distance from the source, the degree of 
spatial coherence grows. The radiation of a laser* has an enormous 
temporal and spatial coherence. At the outlet opening of a laser, 
spatial coherence is observed throughout the entire cross section of 
the light beam. 

It would seem possible to observe interference by passing light 
propagating from an arbitrary source through two slits in an opaque 
screen. With a small spatial coherence of the wave falling on the 
slits, however, the beams of light passing through them will be inco- 
herent, and an interference pattern will be absent. The English 
scientist Thomas Young (1773-1829) in 1802 obtained interference 
from two slits by increasing the spatial coherence of the light falling 
on the slits. Young achieved such an increase by first passing the 
light through a small aperture in an opaque screen. This light was 
used to illuminate the slits in a second opaque screen. Thus, for the 
first time in history, Young observed the interference of light waves 
and determined the lengths of these waves. 


17.3. Ways of Observing the Interference 
of Light 

Let us consider two concrete interference layouts of which one 
uses reflection for splitting a light wave into two parts, and the other 
refraction of light. 

Fresnel’s Double Mirror. Two plane contacting mirrors OM and 
ON are arranged so that their reflecting surfaces form an obtuse angle 
close to ji (Fig. 17.8). Hence, the angle <p in the figure is very small. 
A straight light source S (for example, a narrow luminous slit) is 
placed parallel to the line of intersection of the mirrors O (perpen- 
dicular to the plane of the drawing) at a distance r from it. The mir- 
rors reflect two cylindrical coherent waves onto screen Sc. They 
propagate as if they were emitted by virtual sources S t and S 2 . 
Opaque screen Sc A prevents the direct propagation of the light from 
source S to screen Sc. 

Ray OQ is the reflection of ray SO from mirror OM , and ray OP 
is the reflection of ray SO from mirror ON. It is easy to see that the 

* Lasers will be treated in Vol. Ill of our course. 



362 


Optic $ 


angle between rays OP and OQ is 2<p. Since S and S 1 are symmetrical 
relative to 0M y the length of segment 0S t equals OS , i.e. r. Similar 
reasoning leads to the same result for segment 0S 7 . Thus, the distance 
between sources S x and S 2 is 

d as 2r sin q> « 2 rq> 

Inspection of Fig. 17.8 shows that a = r cos q> « r. Hence, 

I = r -f- 6 


where 6 is the distance from the line of intersection of the mirrors 0 
to screen Sc. 

Using the values of d and l we have found in Eq. (17.10), we obtain 
the width of an interference fringe 


Ax = 


+ b 
2 rq> 


(17.28) 


The region of wave overlapping PQ has a length of 26 tan <p « 26<p. 
Dividing this length by the width of a fringe Ax, we find the maxi* 



mum number of interference fringes that can be observed with the 
aid of Fresnel’s double mirror at the given parameters of a layout: 


M r+bj 


(17.29) 


For all these fringes to be visible indeed, it is essential that N/2 
be not greater than me x tr determined by expression (17.22). 

Fresnel’s Biprism. Two prisms with a small refracting angle 0 
made from a single piece of glass have one common face (Fig. 17.9). 
A straight light source S is arranged parallel to this face at a distance 
a from it. 

It can be shown that when the refracting angle 0 of the prism is 
very small and the angles of incidence of the rays on the face of the 
prism are not very great, all the rays are deflected by the prism 



Interference of Light 


363 


through a practically identical angle equal to 

9 = (n — 1) 0 

(n is the refractive index of the prism). The angle of incidence of 
the rays on the biprism is not great. Therefore, all the rays are deflect- 
ed by each half of the biprism through the same angle. As a result, 



Fig. 17.9 


two coherent cylindrical waves are formed emerging from virtual 
sources S x and S 2 in the same plane as S. The distance between the 
sources is 

d = 2a sin qp « 2aqp =2 a (n — 1) 0 
The distance from the sources to the screen is 

l =a + b 


We find the width of an interference fringe by Eq. (17.10): 


Ax 


a + b 

2a (n — 1 ) 9 


X 


(17.30) 


The region of overlapping of the waves PQ has the length 
2b tan <p ^ 2 6qp = 2b (n — 1)0 

The maximum number of fringes observed is 

mr 4aft (n — l) 2 Q a 

X(a + 6) 


(17.31) 


17.4. Interference of Light Reflected 
from Thin Plates 

When a light wave falls on a thin transparent plate (or film), re- 
flection occurs from both surfaces of the plate. The result is the pro- 
duction of two light waves that in known conditions can interfere. 


364 


Opiict 


Assume that a plane light wave which can be considered as a paral- 
lel beam of rays falls on a transparent plane-parallel plate (Fig. 17.10). 
The plate reflects upward two parallel beams of light. One of them 
was formed as a result of reflection from the top surface of the plate, 
and the second as a result of reflection from its bottom surface (in 
Fig. 17.10 each of these beams is represented by only one ray). The 



second beam is refracted when it enters the plate and leaves it. 
In addition to these two beams, the plate throws upward beams pro- 
duced as a result of three-, five-fold, etc. reflection from the plate 
surfaces. Owing to their small intensity, however, we shall take 
no account of these beams*. We shall also display no interest in the 
beams passing through the plate. 

The path difference acquired by rays 1 and 2 before they meet at 
point C is 

A = ns 2 — s x (17.32) 

where $ x = length of segment BC 

$2 — total length of segments AO and OC 
n = refractive index of the plate. 

We assume that the refractive index of the medium surrounding 
the plate is unity. A glance at Fig. 17.10 shows that s % =2 b tan 0 2 X 
x sin 0 lf and s 2 = 26/cos 0 2 (here b is the thickness of the plate). 


* At n = 1.5, about 5% of the incident luminous flux is reflected from the 
surface of the plate (see the last paragraph of Sec. 16.3). After two reflections, 
the intensity will be 0.05 X 0.05 or 0.25% of the intensity of the initial beam. 
After three reflections, the relevant figure is 0.05 X 0.05 X 0.05, or 0.0125%, 
which is 1/400 of the intensity of the singly reflected beam. 




Interference of Light 


365 


Using these values in Eq. (17.32), we get 


A = 


2 bn 
cos 0 2 


2b tan 0, sin 9* = 2b 


n* — ft sin 0 a sin 0 t 
n cos 0, 


Substituting sin 0 L for n sin 0 2 and taking into account that 


n cos 0 2 = Y n z — n 2 sin 2 0 f = Y nZ — sin 2 0, 


it is easy to give the equation for A the form 

A = 26 V n 2 — sin 2 0, (17.33) 

When calculating the phase difference 6 between the oscillations 
in rays 1 and 2 , it is necessary, in addition to the optical path differ- 
ence A, to take into account the possibility of a change in the phase 
of the wave upon reflection (see Sec. 16.3). At point A (see Fig. 17.10), 
reflection occurs from the interface between the optically less dense 
medium and the optically denser one. Consequently, the wave phase 
experiences a change by jt. At point 0, reflection occurs from the 
interface between the optically denser medium and the optically less 
dense one so that there is no jump in the phase. Hence, an additional 
phase difference equal to n is produced between rays 1 and 2 . It can 
be taken into account by adding to A (or subtracting from it) half 
a wavelength in a vacuum. The result is 

A = 26 n 2 — sin 2 0 t — — (17.34) 


Thus, when a plane wave falls on the plate, two reflected waves 
are formed, and their path difference is determined by Eq. (17.34). 
Let us determine the conditions in which these waves will be cohe- 
rent and can interfere. We shall consider two cases. 

1* A Plane-Parallel Plate. Both plane reflected waves propagate in 
one direction making an angle equal to the angle of incidence 0* 
with a normal to the plate. These waves can interfere if conditions 
of both temporal and spatial coherence are observed. 

For temporal coherence to take place, the path difference given by 
Eq. (17.34) must not exceed the coherence length equal to A, 2 /AX 
» A,}/AX a [see expression (17.21)]. Consequently, the condition 

•26 )/« 2 — sin 2 e,--^-<-^ 


or 

h ^ Xp (WAX, + 1/2) 

2 Y n* — sin 8 

must be observed. In the obtained relation, we may d isregard 1/2 
in comparison with X 0 /AA, 0 . The expression \ n 1 — sin 2 0 X has a mag- 



366 


Optics 


nitude of the order of unity*. We can therefore write 


b< 


2AXo 


(17.35) 


(the double plate thickness must be less than the coherence length). 

Thus, the reflected waves will be coherent only if the plate thick- 
ness b does not exceed the value determined by expression (17.35). 
Assuming that X 0 = 5000 A and AA, 0 = 20 A , we get the extreme 
value of the thickness equal to 


5000 2 

2X20 


6 X 10 5 A = 0.06 mm 


(17.36) 


Now let us consider the conditions for observance of spatial cohe- 
rence. Let us place screen Sc in the path of the reflected beams 


Sc 



(Fig. 17.11). Rays 1' and 2' arriving at point P' will be at a distance 
p' apart in the incident beam. If this distance does not exceed the 
coherence radius p coh of the incident wave, rays V and 2 9 will be 
coherent and will produce at point P 9 an illumination determined by 
the value of the path difference A corresponding to the angle of inci- 
dence 0|. The other pairs of rays travelling at the same angle 0' 
will produce the same illumination at the other points of the screen. 
The screen will thus be uniformly illuminated (in the particular case 
when A = (m + 1/2) X 0 , the screen will be dark). When the inclination 
of the beam is changed (i.e. when the angle 0 X is changed), the illu- 
mination of the screen will change too. 

A glance at Fig. 17.10 shows that the distance between the incident 
rays 1 and 2 is 

p = 2b tan 0 2 cos 0* = r h sin2e i_ (17.37) 

K 2 1 n 2 — sin* 0 t 

* For n =s 1.5, the magnitude of this expression varies within the limits from 
1.12 (at = nf 2) to 1.5 (at = 0). 



Interference of Light 


367 


If we assume that n =1.5, then for 0 X = 45° we get p = 0.86, and 
for 0 A = 10° we get p = 0.16. For normal incidence .(0j = 0), we have 
p = 0 at any n. 

The coherence radius of sunlight has a value of the order of 0.05 mm 
[see Eq. (17.27)]. At an angle of incidence of 45°, we may assume 
that p « 6. Hence, for interference to occur in these conditions, the 
relation 

6 < 0.05 mm (17.38) 

must be observed [compare with Eq. (17.36)]. For an angle of inci- 
dence of about 10°, spatial coherence will be retained at a plate thick- 
ness not exceeding 0.5 mm. We thus arrive at the conclusion that 


P* P' O 



owing to the restrictions imposed by temporal and spatial coherence, 
interference is observed when a plate is illuminated by sunlight 
only if the thickness of the plate does not exceed a few hundredths 
of a millimetre. Upon illumination with light having a greater de- 
gree of coherence, interference is also observed in reflection from 
thicker plates or films. 

Interference from a plane-parallel plate is observed in practice 
by placing in the path of the reflected beams a lens that gathers the 
rays at one of the points of the screen in the focal plane of the lens 
(Fig. 17.12). The illumination at this point depends on the value of 
quantity (17.34). When A = mX 0 , we get maxima, and when A = 

= + X 0 — minima of the intensity (m is an integer). The con- 

dition for the maximum intensity has the form 

2b Yn? — sin 2 0, = (m + ^-)x 0 (17.39) 

Assume that a thin plane-parallel plate is illuminated by diffuse 
monochromatic light (see Fig. 17.12). Let us arrange a lens parallel 
to the plate and put a screen in the focal plane of the lens. Diffuse 



368 


Optics 


light contains rays of the most diverse directions. The rays parallel 
to the plane of the drawing and falling on the plate at the angle 0' 
after reflection from both surfaces of the plate will be gathered by 
the lens at point P' and will set up at this point an illumination 
determined by the value of the optical path difference. Rays propa- 
gating in other planes but falling on the plate at the same angle 0' 
will be gathered by the lens at other points at the same distance as 
point P 9 from centre O of the screen. The illumination at all these 
points will be the same. Thus, the rays falling on the plate at the 
same angle 0' will produce on the screen a collection of identically 
illuminated points arranged along a circle with its centre at O „ 
Similarly, the rays falling at a different angle 0^ will produce on the 
screen a collection of identically (but different in value because A 
is different) illuminated points arranged along a circle of another 
radius. The result will be the appearance on the screen of a system 
of alternating bright and dark circular fringes with a common centre 
at point O. Each fringe is formed by the rays falling on the plate at 
the same angle 0 X . This is why interference fringes produced in such 
conditions are known as fringes of equal inclination. When the lens 
is arranged differently relative to the plate (the screen must coin- 
cide with the focal plane of the lens in all cases), the fringes of equal 
inclination will have another shape. 

Every point of an interference pattern is due to rays which formed 
a parallel beam before passing through the lens. Hence, in observing 
fringes of equal inclination, the screen must be placed in the focal 
plane of the lens, i.e. in the same way in which it is arranged to pro- 
duce an image of infinitely remote objects on it. Accordingly, fringes 
of equal inclination are said to be localized at infinity. The part of 
the lens can be played by the crystalline lens, and that of the screen 
by the retina of the eye. In this case for observing fringes of equal 
inclination, the eye must be accommodated as when looking at very 
remote objects. 

According to Eq. (17.39), the position of the maxima depends on 
the wavelength X 0 . Therefore, in white light, we get a collection of 
fringes displaced relative to one another and formed by rays of differ- 
ent colours; the interference pattern acquires the colouring of a rain- 
bow. The possibility of observing an interference pattern in white 
light is determined by the ability of the eye to distinguish light 
tints of close wavelengths. The average human eye perceives rays 
differing in wavelength by less than 20 A as having the same colour. 
Therefore, to assess the conditions in which interference from plates 
can be observed in white light, we must assume that AK 0 equals 20 A. 
We took exactly this value in assessing the thickness of a plate [see 
Eq. (17.36)]. 

2. Plate of Varying Thickness. Let us take a plate in the form of 
a wed?e with an aoex anffle of w (Ficr. 17.13). Assume that a parallel 



Interference of Light 


369 


beam of rays falls on it. Now the rays reflected from different surfaces 
of the plate will not be parallel. Two rays that practically merge 
before falling on the plate (in Fig. 17.13 they are depicted in the form 
of a single straight line designated by the figure 1') intersect after 
reflection at point Q The two rays 1 ” practically merging intersect 
at point Q” after reflection. It can be shown that points Q\ Q* and 
other points similar to them lie in one plane passing through apex 0 of 
the wedge. Ray V reflected from the bottom surface of the wedge and 



ray 2' reflected from its top surface will intersect at point R ' that is 
closer to the wedge than Q ' . Similar rays V and 3 f will intersect at 
point P' that is farther from the wedge surface than Q'. 

The directions of propagation of the waves reflected from the top 
and bottom surfaces of the wedge do not coincide. Temporal cohe- 
rence will be observed only for the parts of the waves reflected from 
places of the wedge for which the thickness satisfies condition (17.35). 
Assume that this condition is observed for the entire wedge. In addi- 
tion, assume that the coherence radius is much greater than the wedge 
length. Hence, the reflected waves will be coherent in the entire space 
over the wedge, and no matter at what distance from the wedge the 
screen is, an interference pattern will be observed on it in the form 
of fringes parallel to the wedge apex O (see the last three paragraphs 
of Sec. 17.1). This, particularly, is how matters are when a wedge is 
illuminated by light emitted by a laser. 

With restricted spatial coherence, the region of localization of the 
interference pattern (i.e. the region of space in which an interference 
pattern can be seen on a screen placed in it) will be restricted too. 
If we arrange a screen so that it passes through points Q\ Q 0 , . . . 
(see screen Sc in Fig. 17.13), an interference pattern will appear on it 
even if the spatial coherence of the falling wave is extremely small 
(rays that coincided before falling on the wedge will intersect at 
points on the screen). At a small wedge angle <p, the path difference 



370 


Optics 


of the rays can be calculated with sufficient ac uracy by Eq. (17.34) t 
taking as b the thickness of the plate at the place where the rays fall 
on it. Since the path difference for the rays reflected from different 
sections of the wedge is now different, the illumination of the screen 
will be non-uniform— bright and dark fringes will appear on it (see 
the dash curve showing the illumination of screen Sc in Fig. 17.13). 



Each of these fringes is produced as a result of reflection from sections 
of the wedge having the same thickness. This is why they are known 
as fringes of equal thickness. 

Upon displacement of the screen from position Sc in a direction 
away from the wedge or toward it, the degree of spatial coherence of 
the incident wave begins to tell. If in the position of the screen denot- 
ed in Fig. 17.13 by Sc\ the distance p' between the incident rays V 
and 2' becomes of the order of the coherence radius, no interference 
pattern will be observed on screen Sc'. Similarly, the pattern van- 
ishes when the screen is at position Sc". 

Thus, the interference pattern produced when a plane wave is 
reflected from a wedge is localized in a certain legion near the surface 
of the wedge. This region becomes narrower when the degree of spa- 
tial coherence of the incident wave diminishes. Inspection of Fig. 1*7.13 
shows that the conditions for both temporal and spatial coherence 
become more favourable nearer to the apex of the wedge. Therefore, 
the distinctness of the interference pattern diminishes when moving 
from the apex of the wedge to its base. A pattern may be observed 
only for the thinner part of the wedge. For its remaining part, the 
screen will be uniformly illuminated. 

Practically, fringes of equal thickness are observed by placing 
a lens near a wedge, and a screen behind the lens (Fig. 17.14). The 
part of the lens can be played by the crystalline lens, and of the screen 


Interference of Light 


371 


by the retina of the eye. If the screen behind the lens is in a plane 
conjugated with the plane designated by Sc in Fig. 17.13 (the eye is 
accordingly accommodated to this plane), the pattern will be most 
distinct. When the screen onto which the image is projected is moved 
(or when the lens is moved), the pattern will become less distinct and 
will vanish completely if the plane conjugated with the screen passes 
beyond the limits of the region of localization of the interference 
pattern observed without a lens. 

When observed in white light, the fringes will be coloured, so that 
the surface of a plate or film will have rainbow colouring. For exam- 
ple, thin films of oil on the surface of wat- 
er and soap films have such colouring. 

The temper colours appearing on the sur- 
face of steel articles when they are hard- 
ened are also due to interference from a 
film of transparent oxides. 

Let us compare the two cases of inter- 
ference upon reflection from thin films 
which we have considered. Fringes of 
equal inclination are obtained when a 
plate of constant thickness ( b = const) is 
illuminated by diffuse light containing 
rays of various directions (0! is varied with- 
in more or less broad limits). Fringes of 
equal inclination are localized at infinity. Fig. 17.15 

Fringes of equal thickness are observed 

when a plate of varying thickness ( b varies) is illuminated by a parallel 
beam of light (0J = const). Fringes of equal thickness are localized 
near the plate. In real conditions, for example, when observing rain- 
bow colours on a soap or oil film, both the angle of incidence of the 
rays and the thickness of the film are varied. In this case, fringes of 
a mixed type are observed. 

We must note that interference from thin films can be observed 
not only in reflected, but also in transmitted light. 

Newton’s Rings. A classical example of fringes of equal thickness 
are Newton’s rings. They are observed when light is ’•effected from 
a thick plane-parallel glass plate in contact with a plano-convex 
lens having a large radius of curvature (Fig, 17.15). The part of a 
thin film from whose surfaces coherent waves are reflected is played 
by the air gap between the plate and the lens (owing to the great thick- 
ness of the plate and the lens, no interference fringes appear as a re- 
sult of reflections from other surfaces). With normal incidence of the 
light, fringes of equal thickness have the form of concentric rings, 
and with inclined incidence, of ellipses. Let us find the radii of New- 
ton’s rings produced when light falls along a normal to the plate. In 
this case, sin 0 a = 0, and the optical path difference equals the double 






372 


Optics 


thickness of the gap [see Eq. (17.33), it is assumed that n = 1 in 
the gap). It follows from Fig. 17.15 that 

R* = (if _ 6)* + r» « - 2 Rb + r* (17.40) 

where R — radius of curvature of the lens 

r = radius of a circle with the identical gap b corresponding 
to all of its points. 

Owing to the smallness of b, in expression (17.40) we have disre- 
garded the quantity 6* in comparison with 2Rb. In accordance with 
expression (17.40), b = rV2R. To take account of the change in 
the phase by n occurring upon reflection from the plate, we must add 
X 0 /2 to 2b = rVR. The result is 

A = -s-+T- (17.41) 

At points for which A = m"k 0 = 2 m' (X 0 /2), maxima appear, 
and at points for which A= (m' j A, 0 = (2m' -f 1) (X 0 /2), mini- 
ma of the intensity. Both conditions can be combined into the single 
one 

A = m-1T 

maxima corresponding to even values of to, and minima of the inten- 
sity, to odd values. Introducing into this expression Eq. (17.41) 
for A and solving the resulting equation relative to r, we find the 
radii of bright and dark Newton’s rings: 

r =’ ( m = 1,2,3,...) (17.42) 

Radii of bright rings correspond to even to’ s, and radii of dark rings 
to odd ones. The value r = 0 corresponds to to = 1, i.e. to the point 
at the place of contact of the plate and the lens. A minimum of inten- 
sity is observed at this point. It is due to the change in the phase by 
n when a light wave is reflected from the plate. 

Coating of Lenses. The coating of lenses is based on the interfer- 
ence of light when reflected from thin films. The transmission of 
light through each refracting surface of a lens is attended by the re- 
flection of about four per cent of the incident light. In multicompo- 
nent lenses, such reflections occur many times, and the total loss 
of the light flux reaches an appreciable value. In addition, the re- 
flections from the lens surfaces result in the appearance of highlights* 
The reflection of light is eliminated by applying a thin film of a sub- 
stance having a refractive index other than that of the lens to each 
free surface of the latter. The components obtained in this way are 
called coated lenses. The thickness of the coating is chosen so that 



Interference of Light 


373 


the waves reflected from both its surfaces interfere destructively. 
An especially good result is obtained if the refractive index of the 
film equals the square root of the refractive index of the lens. When 
this condition is satisfied, the intensity of both waves reflected from 
the film surfaces is the same. 


17.5. The Michelson Interferometer 



Many varieties of interference instruments called interferometers 
are in use. Figure 17.16 is a schematic view of a Michelson interfero- 
meter*. A light beam from source S falls on semitransparent plate 
P x coated with a thin layer of silver (this 
layer is depicted by dots in the figure). 

Half of the incident light flux is reflect- 
ed by plate P 1 in the direction of ray 7. 
and half passes through the plate and pro- 
pagates in the direction of ray 2 . Beam 7 
is reflected from mirror M x and returns to 
P x , where it is split into two beams of 
equal intensity. One of them passes 
through the plate and forms beam 7', and 
the second one is reflected in the direction 
of 5. The latter beam will no longer inter- 
est us. Beam 2 after being reflected by 
mirror M 2 also returns to plate P x where 
it is divided into two parts: beam 2' re- 
flected from the semitransparent layer, 

and the beam transmitted through the layer, which will also no lon- 
ger interest us. Light beams T and 2' have the same intensity. 

If conditions of temporal and spatial coherence are observed, 
beams 7' and 2' will interfere. The result of this interference depends 
on the optical path difference from plate P x to mirrors M x and M % 
and back. Ray 2 passes through the plate three times, and ray 7 
only once. To compensate the resulting change in the optical path 
difference (owing to dispersion) for waves of different lengths, plate P 2 
is placed in the path of ray 7. Plates P x and P 2 are identical, except 
for the silver coating on the former. This arrangement makes the 
paths of rays 7 and 2 in glass equal. The interference pattern is ob- 
served with the aid of telescope T . 

Let us mentally replace mirror Af 2 with its virtual image Af' in 
semitransparent plate P x . Beams 7' and 2' can thus be considered as 
due to reflection from a transparent plate contained between planes 
M x and Af'. We can use adjusting screws W x to change the angle 

* Named after its inventor, the American physicist Albert Michelson (1852- 
1931). 




374 


Optics 


between these planes; in particular, they can be arranged strictly 
parallel to each other. By rotating micrometric screw W tf we can 
smoothly move mirror M t without changing its inclination. We can 
thus change the thickness of the "plate”; in particular, we can make 
planes M x and M\ intersect (Fig. 17.166). 

The nature of the interference pattern depends on the adjustment 
of the mirrors and on the divergence of the beam of light falling on 
the instrument. If the beam is parallel, and planes M 1 and M' t make 
an angle other than zero, then straight fringes of equal thickness paral- 
lel to the lines of intersection of planes M x and M' will be observed 
in the field of vision of the telescope. In white light, all the fringes 
except the one coinciding with the line of intersection of the zero- 
order fringe will be coloured. The zero-order fringe will be black be- 
cause beam 1 is reflected from plate P 1 from the outside, and beam 2 
from the inside. As a result, a phase difference equal to n is produced 
between them. In white light, fringes are observed only with a small 
thickness of “plate” M,M' 2 [see Eq. (17.36)1. In monochromatic light 
corresponding to the red line of cadmium, Michelson observed a dis- 
tinct interference pattern at a path difference of the order of 500 000 
wavelengths (the distance between M x and M' t in this case is about 
150 mm). 

With a slightly diverging beam of light and a strictly parallel ar- 
rangement of planes M x and M', fringes of equal inclination are 
obtained that have the form of concentric rings. When micrometric 
screw W t is rotated, the diameter of the rings grows or diminishes. 
Either new rings appear at the centre of the pattern, or the dimin- 
ishing rings shrink to a point and then vanish. Displacement of 
the pattern by one fringe corresponds to movement of mirror M t 
through half a wavelength. 

Michelson used the instrument described above to carry out several 
experiments that entered the annals of physics. The most famous of 
them, performed together with the American chemist Edward Morley 
(1838-1923) in 1887, had the aim of detecting motion of the Earth 
relative to the hypothetic ether (we shall treat this experiment in 
Sec. 21.3). In 1890-1895, Michelson used the interferometer he had 
invented to make the first comparison of the wavelength of the red 
line of cadmium with the length of the standard metre. 

In 1920, Michelson constructed a stellar interferometer which he 
used to measure the angular dimensions of stars. This instrument 
was mounted on a telescope. A screen with two slits was installed 
in front of the objective of the telescope (Fig. 17.17). The light from 
a star was reflected from a symmetrical system of mirrors M lt Af„ 
M 3 , and M k installed on a rigid frame fastened on a carriage. The 
inner mirrors Af, and M 4 were fixed, and the outer ones and M t 
could move symmetrically away from or toward mirrors M, and M 4 . 
The path of the rays is clear from the figure. Interference fringes were 



Interference of Light 


375 


produced in the focal plane of the telescope objective. Their visibil- 
ity* depended on the distance between the outer mirrors. By moving 
these mirrors, Michelson determined the distance l between them at 
which the visibility of the fringes vanishes. This distance must be 
of the order of the coherence radius of a light wave arriving from 



a star. According to expression (17.26), the coherence radius is X/<p. 
The condition l = X/cp gives the angular diameter of a star 



Accurate calculations give the formula 

9=^4- 

where A ~ 1.22 for a source in the form of a uniformly illuminated 
disk. If the disk is darker at its edges than at the centre, the coef- 
ficient exceeds 1.22, its value depending on the rate of diminishing 
of the illumination in the direction from the centre toward the edge. 
In addition, accurate calculations show that after vanishing at a 
certain value of I, the visibility upon a further increase in l again 
becomes other than zero; however, the values it reaches are not great. 

The maximum distance between the outer mirrors in the stellar 
interferometer constructed by Michelson was 6.1 m (the diameter 
of the telescope was 2.5 m). A minimum measurable angular diameter 
of about 0.02' corresponded to this distance. The first star whose 
angular diameter was measured was Betelgeuse (alpha Orion). The 
value of <p obtained for it was 0.047'. 

* The visibility of a fringe is defined as the quantity 

y — fjnax — /min 
7 max "f* /min 

where /max and /min are the maximum and minimum intensities of the light 
in the vicinity of the given fringe, respectively. 



376 


Optiet 


17 . 6 . Multibeam Interference 


Up to now, we have dealt with two-beam interference. Now let us 
investigate the interference of many light rays. 

Assume that N rays of the same intensity arrive at a given point 
of a screen, the phase of each following ray being shifted relative 
to that of the preceding one by the same value 6. Let us represent 
the oscillations set up by the rays in the form of exponents: 

£, = ae ia “; E t = ae* . . . , E m = ae* !•*+(«•- 

= 1>«] 


where a is the amplitude of an oscillation. The resultant oscillation 
is determined by the formula 

Jf N 

2 E m = ae*** 2 e* <*-»>' 6 

m— 1 m=l 

The expression obtained is the sum of N terms of a geometrical pro- 
gression with its first term equal to unity and its common ratio equal 
to e {0 . Hence, 

E = ae*** - - .. = Ae*** 

i—e * 6 

where 

4 JN6 

A = al fzjr < 17 * 43 > 

is the complex amplitude that can be represented in the form 

A = Ae ia (17.44) 


( A is the usual amplitude of the resultant oscillation, and a is its 
initial phase). 

The product of quantity (17.44) and its complex conjugate gives 
the square of the amplitude of the resultant oscillation: 

AA* = Ae ia Ae~ ia A 2 (17 45) 


Substituting for .4 in Eq. (17.45) its value from Eq. (17.43), we get 
the following expression for the square of the amplitude: 


A 2 = AA* = a 2 


(j — e***) ( 1 - 




(i-e i6 ) (1 — e~ i0 ) 


- 10 \ 


= a* 


2-e iWd - g 

2—e i6 —e~ i6 


plitude: 

— i Nb 


= a 2 


1 — cos N 6 
1 — cos 6 


sin 1 (N6/2) 
sin 1 (6/2) 


(17.46) 


The intensity is proportional to the square of the amplitude. Hence, 
the intensity produced upon the interference of the N rays being con- 



Interference of Light 


377 


sidered is determined by the expression 


I 


(6) = Ka 2 


sin* (7V6/2) _ r sin* (N6/2) 
sin* (6/2) — y ° sin* (6/2) 


(17.47) 


(* is a constant of proportionality, I 0 = Aa* is the intensity pro- 
duced by each of the rays separately). 

At the values 


6 = 2nm (in = 0, ±1, ±2, . . .) (17.48) 

Eq. (17.47) becomes indeterminate. For this reason, we apply L’Hos- 
pital’s rule: 

sin* (Nd/2) 2 sin (Nd/2) cos (N6/2)-N/2 ,. M sin (/V6) 

6 sin* (6/2) 2 sin (6/2) cos (6/2). 1/2 sin 6 


The expression obtained is also indeterminate. For this reason, we 
apply L’Hospital’s rule again: 


lim 

6-+2nm 


sin 8 (/V6/2) 
sin 8 (6/2) 


lim 

6-+2nm 


jy sin (/V6) _ 
sin 6 


lim 

G— 2nm 


v N cos (/V6) _ 
cos 6 


Thus, when 6 = 2 nm (or when the path differences A = mX 0 ), 
the resultant intensity is 

I = I<>N* (17.49) 

This result could have been predicted. Indeed, all the oscillations 
arrive at points for which 6 = 2 nm in the same phase. Hence, the 
resultant amplitude is N times the amplitude of a separate oscilla- 
tion, and the intensity is N 2 times that of a separate oscillation. 

Let us call the spots where the intensity determined by Eq. (17.49) 
is observed the principal maxima. Their position is determined by 
condition (17.48). The number m is called the order of the principal 
maximum. It can be seen from Eq. (17.47) that the space between 
two adjacent principal maxima accommodates N — 1 minima of the 
intensity. To verify this statement, let us consider, for example, the 
interval between the maxima of the zero (m = 0) and of the first 
(m = 1) order. In this interval, 6 changes from zero to 2 ji, and 6/2 
from zero to n. The denominator of Eq. (17.47) is other than zero 
everywhere except for the ends of the interval. It reaches its maximum 
value equal to unity at the middle of the interval. The quantity 
N6/2 takes on all the values from zero to Nn within the interval being 
considered. At values of n, 2jt, . . ., (Af — 1) Jt, the numerator of 
Eq. (17.47) becomes equal to zero. Here we have minima of the inten- 
sity. Their positions correspond to values of 6 equal to 

6 = ^-2 n (*' = 1,2, ...,A-1) (17.50) 


There are N — 2 secondary maxima in the intervals between the 
N — 1 minima. The secondary maxima closest to the principal maxi- 



378 


Optics 


ma have the greatest intensity. The secondary maximum closest to 
the principal zero-order maximum is between the first ( k ' =1) and 
second (k' = 2) minima. Values of 8 equal to 2 ntN and 4 n!N cor- 
respond to these minima. Hence, 6 = 3n!N corresponds to the secon- 
dary maximum being considered. Introduction of this value into 
Eq. (17.47) yields 


/ (3ji/7V) = Ka 2 


sin* (3n/2) 
sin* (3n/2N) 


The numerator equals unity. At a great value of V, we may assume 
that the sine in the denominator equals its argument Isin ( 3n/2N ) & 
^ 3n/2N). Hence, 


/ (3 n/N) ^ Ka 2 


i 

(3n/2W)» 


Ka % N* 

(3n/2)* 


The quantity in the numerator is the intensity of the principal maxi- 
mum [see Eq. (17.49)]. Thus, at a great value of N> the secondary 



maximum closest to the principal maximum has an intensity that 
is 1/(3 jt/ 2) 2 ^ 1/22 of the intensity of the principal maximum. The 
other secondary maxima are still weaker. 

Figure 17.18 shows a plot of the function / (6) for N = 10. For 
comparison, a plot of the intensity for N = 2 [two-beam interfer- 
ence; see the curve / ( x ) in Fig. 17.21 is shown by a dash line. In- 
spection of the figure shows that the principal maxima become nar- 
rower and narrower with an increase in the number of interfering 
rays. The secondary maxima are so weak that the interference pat- 
tern practically has the form of narrow bright lines on a dark back- 
ground. 

Now let us consider the interference of a very great number of 
rays whose intensity diminishes in a geometrical progression. The 



Interference of Light 


379 


oscillations being added have the form 

E x = ae iat , E , = ape* .... 

E m = ap m ~ 1 e i t »*+(»- d fi], ... (17.51) 

(p is a constant quantity less than unity). The resultant oscillation 
is described by the equation 


N N 

E = 2 E m — ae iot 2 <»-*>• 

m=*» 1 m~i 


Using the expression for the sum of the terms of a geometrical pro- 
gression, we get 


E = ae i(at 


4 -NJN6 
a — ■ p e 


1 — pe 


id 


Ae'M 


Thus, the complex amplitude is 


A - a ~T^- <i7 - 52) 

If N is very great, the complex number p N e im may be disregarded 
in comparison with unity (we shall indicate as an example that 
0.9 100 «4 X 10~ 4 ). Equation (17.52) is thus simplified as follows: 


Multiplying this equation by its complex conjugate, we get the 
square of the ordinary amplitude of the resultant oscillation: 


A 2 = AA *: 


(1 — pe i6 ) ( 1 - pe “ ' iG ) 1 -f p* — p (e i6 + e “ w ) 


1 + p* — 2p cos 6 (1 — p)* + 2p (1 — cos 6) (1 — p) a H- 4p sin* (6/2) 


Hence, 


'W-TIHo* 


Ka* 


(1 — P)*+4p sin* ( 6 / 2 ) (1 — p)*+4p sin* (6/2) 


(17.53) 


where /, = Ka 2 is the intensity of the first (most intensive) ray. 
At values of 

6 = 2nm (to = 0, ±1, ±2, . . .) (17.54) 

Eq. (17.53) has maxima equal to 

/ lx 

1 max (1— p)* 


(17.55} 



380 


0ptl€9 


In the intervals between maxima, the function changes monotonously, 
reaching a value equal to 


h _ h 

mm ~ (1— p)*+4p ~ (l + p) a 


(17.56) 


at the middle of the interval. Thus, the ratio of the intensity at a 
maximum to that at a minimum 


^mai / l~hP \ 2 

Jmln l 1 — P ' 


(17.57) 


is the greater, the closer p is to unity, i.e. the slower is the rate of 
diminishing of the intensity of the interfering rays. Figure 17.19 



Fig. 17.19 


shows a graph of function (17.53) for p = 0.8. It can be seen from 
the figure that the interference pattern has the form of narrow sharp 

lines on a virtually dark background. 
Unlike Fig. 17.18, secondary maxi- 
ma are absent. 

A practical case of a great num- 
P ber of rays with a diminishing in- 
tensity is encountered in the Fabry- 
Perot interferometer. This instru- 
ment consists of two glass or quarts 
plates separated by an air gap 
Fig. 17.20 (Fig. 17.20). The internal surfaces 

of the plates are thoroughly pol- 
ished so that the irregularities on them do not exceed several hun- 
dredths of the length of a light wave. Next partly transparent metal 
layers or dielectric films* are applied to these surfaces. The outer 

* Metal layers have the shortcoming that they absorb light rays to a great 
extent. This is why recent years have seen their replacement with multilayer di- 
electric coatings having a high reflectivity. 




Interference of Light 


381 


surfaces of the plates are at a slight angle relative to the inner ones to 
eliminate the highlights due to the reflection of light from these 
surfaces. In the original design of the interferometer, one of the 
plates could be moved relative to the other stationary one with 
the aid of a micrometric screw. The unreliability of this design, 
however, resulted in its coming out of use. In modern designs, the 
plates are secured rigidly. The parallelity of the internal working 



planes is achieved by installing an invar or quartz ring* between 
the plates. This ring has three projections with thoroughly polished 
edges at each side. The plates are pressed against the ring by springs. 
This design reliably ensures strict parallelity of the internal planes 
of the plates and constancy of the distance between them. Such an 
interferometer with a fixed distance between its plates is known as a 
Fabry-Perot etalon. 

Let us see what happens to a ray entering the gap between the 
plates (Fig. 17.21). Assume that the intensity of the entering ray is 
/ 0 . At point A x , this ray is divided into ray 1 emerging outward 
and reflected ray 1 '. If the coefficient of reflection from the surface 
of the plate is p, then the intensity of ray 1 will be I x = (1 — p) / 0 , 
and the intensity of the reflected ray will be /' = p/ 0 **. At point fl,, 


* Both these materials are distinguished by their extremely low tempera- 
ture coefficient of expansion. 

** We disregard the absorption of light in the reflecting layers and inside 
the plates. 



382 


Optics 


ray V is divided into two. Ray 1 " shown by a dash line will drop 
out of consideration, while reflected ray 1 " will have an intensity 
of /' = p/j = p 2 / 0 . At point A 2 , ray 1" will be divided into two 
rays — ray 2 emerging outward having an intensity of I 2 = (1 — 
— p) /' = (1 — p) p 2 / 0 and reflected ray 2 ', and so on. Thus, the 
following relation holds for the intensities of rays 1, 2, 3, etc. emer- 
ging from the instrument: 

A : ^ 3 : 1 3 • . . . = 1 : p 2 : p 4 : ... 

Accordingly, for the amplitudes of the oscillations we have 
A t : A 2 : A 3 : . . . = 1 : p : p 2 : ... 

[compare with Eq. (17.51)1. 

The oscillation in each of the rays 2,3.4, ... lags in phase behind 
the oscillation in the preceding ray by the same amount 6 determined 



Fiij. 17.22 

by the optical path difference A appearing on the path Ax-ZVA* 
or A t -B 2 -A 3 , etc. (see Fig. 17.21). A glance at the figure shows that 
A = 2Z/cos <p, where <p is the angle of incidence of the rays on the 
reflecting layers. 

If we gather rays 1,2,3 , ... with the aid of a lens at point P 
of its focal plane (see Fig. 17.20), then the oscillations produced by 
these rays will have the form given by Eq. (17.51). Hence, the inten- 
sity at point P is determined by Eq. (17.53), in which p has the mean- 





Interference of Light 


383 


ing of the coefficient of reflection, and 

c 2n 21 

X cos <p 

When a diverging beam of light is passed through the instrument, 
fringes of equal inclination having the form of sharp rings (Fig. 17.22) 
will be produced in the focal plane of the lens. 

The Fabry-Perot interferometer is used in spectroscopy to study 
the fine structure of spectral lines. It has also come into great favour 
in metrology for comparing the length of the standard metre with 
the wavelengths of individual spectral lines. 



CHAPTER 18 DIFFRACTION OF LIGHT 


18.1, Introduction 

By diffraction is meant the combination of phenomena observed 
when light propagates in a medium with sharp heterogeneities* and 
associated with deviations from the laws of geometrical optics. Dif- 
fraction, in particular, leads to light* waves bending around obstacles 
and to the penetration of light into the region of a geometrical sha- 
dow. The bending of sound waves around obstacles (i.e. the diffrac- 
tion of sound waves) is constantly observed in our everyday life. 
To observe the diffraction of light waves, special conditions must 
be set up. This is due to the smallness of the lengths of light waves. 
We know that in the limit, when X — 0, the laws of wave optics 
transform into those of geometrical optics. Hence, other conditions 
being equal, the deviations from the laws of geometrical optics de- 
crease with a diminishing wavelength. 

There is no appreciable physical difference between interference 
and diffraction. Both phenomena consist in the redistribution of the 
light flux as a result of superposition of the waves. For historical 
reasons, the redistribution of the intensity produced as a result of 
the superposition of waves emitted by a finite number of discrete 
coherent sources has been called the interference of waves. The redis- 
tribution of the intensity produced as a result of the superposition of 
waves emitted by coherent sources arranged continuously has been 
called the diffraction of waves. We therefore speak about the inter- 
ference pattern from two narrow slits and about the diffraction pat- 
tern from one slit. 

Diffraction is usually observed by means of the following set-up. 
An opaque barrier closing part of the wave surface of the light wave 
is placed in the path of a light wave propagating from a certain source. 
A screen on which the diffraction pattern appears is placed after the 
barrier. 

Two kinds of diffraction are distinguished. If the light source S 
and the point of observation P are so far from a barrier that the rays 
falling on the barrier and those travelling to point P form virtually 
parallel beams, we have to do with diffraction in parallel rays or 
with Fraunhofer diffraction. Otherwise, we have to do with Fresnel 


*i For example, near the boundaries of opaque or transparent bodies, through 



Diffraction of Light 


385 


diffraction. Fraunhofer diffraction can be observed by placing a lens 
after light source S and another one in front of point of observation P 
so that points S and P lie in the focal plane of the relevant lens 
(Fig. 18.1). 



The criterion allowing us to determine the kind of diffraction we 
are dealing with — Fresnel or Fraunhofer — in each specific case will 
be given in Sec. 18.5. 


18.2. Huygens-Fresnel Principle 

The penetration of light waves into the region of a geometrical 
shadow can be explained with the aid of Huygens’ principle (see 
Sec. 16.9). This principle, however, gives no 
information on the amplitude and, consequent- 
ly, on the intensity of waves propagating in 
different directions. The French physicist 
Augustin Fresnel (1788-1827) supplemented 
Huygens’ principle with the concept of the inter- 
ference of secondary waves. Taking into ac- 
count the amplitudes anci phases of the secon- 
dary waves makes it possible to find the ampli- 
tude of the resultant wave for any point of 
space. Huygens’ principle developed in this 
way was named the Huygens-Fresnel principle. 

According to the Huygens-Fresnel princi- 
ple, every element of wave surface S (Fig. 18.2) 
is the source of a secondary spherical wave whose amplitude is pro- 
portional to the size of element dS . The amplitude of a spherical wave 
diminishes with the distance r from its source according to the law 
1/r [see Eq. (14.12)1. Consequently, the oscillation 

dE = K cos (a it — kr + a 0 ) 



(18.1) 



386 


Optics 


arrives from each section dS of a wave surface at point P in front of 
this surface. In Eq. (18.1), (cat + a 0 ) is the phase of the oscillation 
where wave surface S is, k is the wave number, r is the distance from 
surface element dS to point P. The factor a 0 is determined by the 
amplitude of the light oscillation at the location of dS. The coefficient 
K depends on the angle <p between a normal n to area dS and the 
direction from dS to point P. When cp = 0, this coefficient is maxi- 
mum; when q> = n/2, it vanishes. 

Sc The resultant oscillation at point P is the 

' | j superposition of the oscillations given by Eq. 

! j j (18.1) taken for the entire wave surface S : 

j j L ./> Ewm f K(<?)-^rCos(a>t— Jcr + a 0 )dS (18.2) 

j j j s 

i ! | This equation is an analytical expression of 

l * ! the Huygens-Fresnel principle. 

The Huygens-Fresnel principle can be sub- 
Fig. 18.3 stantiated by the following reasoning. Assume 

that thin opaque screen Sc (Fig. 18.3) is 
placed in the path of a light wave (we shall consider it plane for sim- 
plicity’s sake). The intensity of the light everywhere after the screen 
will be zero. The reason is that the light wave falling on the screen 
produces oscillations of the electrons in the material of the screen. 
The oscillating electrons emit electromagnetic waves. The field after 
the screen is a superposition of the primary wave (falling on the 
screen) and all the secondary waves. The amplitudes and phases 
of the secondary waves are such that upon superposition of these 
waves with the primary one, a zero amplitude is obtained at any point 
P after the screen. Consequently, if the primary wave produces the 
oscillation 

A pnm COS((i)t + a) 

at point P, then the resultant oscillation produced by the secondary 
waves at the same point has the form 

_4 gec cos (<*>* + a — n) 

Here A sec = Ap rIrn . 

What has been said above signifies that when calculating the ampli- 
tude of an oscillation set up at point P by a light wave propagating 
from a real source, we can replace this source with a collection of 
secondary sources arranged along the wave surface. This is exactly 
the essence of Huygens-Fresnel principle. 

Let us divide the opaque barrier into two parts. One of them, which 
we shall call a stopper, has finite dimensions and an arbitrary shape 
(a circle, rectangle, etc.). The other part includes the entire remain- 
ing surface of the infinite barrier. As long as the stopper is in place, 
the resultant oscillation at point P after the barrier is zero. It can be 


Diffraction of Light 


387 


represented as the sum of the oscillations set up by the primary 
wave, the wave produced by the stopper, and the wave produced by 
the remaining part of the barrier: 

>lprim cos (c o* -f- a) + A slop cos (co* + a') -r 

+ A bar COS (co* + a") = 0 (18.3) 

If the stopper is removed, i.e. the wave is transmitted through 
the apreture in the opaque barrier, then the oscillation at point P 
will have the form 

E P = A prlm cos (<d* -f a) -j- A bar cos (a>* + a") = 

= — ^4stop cos (CD* + a') = A stoP cos (c o* + a' — n) 

We have used condition (18.3) and assumed that removal of the stop- 
per does not change the nature of the oscillations of the electrons 
in the remaining part of the barrier. 

We can thus consider that the oscillations at point P are produced 
by a collection of sources of secondary waves on the surface of the 
aperture formed after removal of the stopper. 

18.3. Fresnel Zones 

The performance of calculations by Eq. (18.2) is a very difficult 
task in the general case. As Fresnel showed, however, the amplitude 
of the resultant oscillation can be found by simple algebraic or geo- 
metrical summation in cases distinguished by symmetry. 



To understand the essence of the method developed by Fresnel, 
let us determine the amplitude of the light oscillation set up at 
point P by a spherical wave propagating in an isotropic homogeneous 
medium from point source S (Fig. 18.4). The wave surfaces of such 



388 


Optics 


waves are symmetrical relative to straight line SP . Taking advantage 
of this circumstance, let us divide the wave surface shown in the 
figure into annular zones constructed so that the distances from the 
edges of each zone to point P differ by X/2 (X is the length of the wave 
in the medium in which it is propagating). Zones having this 
property are known as Fresnel zones. 

A glance at Fig. 18.4 shows that the distance b m from the outer 
edge of the m-th zone to point P is 

b m = b + m± (18.4) 

(b is the distance from the crest O of the wave surface to point P). 

The oscillations arriving at point P from similar points of two 
adjacent zones (i.e. from points at the middle of the zones, or at the 



outer edges of the zones, etc.) are in counterphase. Therefore, the 
resultant oscillations produced by each of the zones as a whole will 
differ in phase for adjacent zones by jx too. 

Let us calculate the areas of the zones. The outer boundary of the 
m-th zone separates a spherical segment of height h m on the wave 
surface (Fig. 18.5). Let the area of this segment be S m , Hence, the 
area of the m-th zone can be written as 

AS m = — ^m-l 

where is the area of the spherical segment separated by the outer 
boundary of the (m — l)-th zone. 

It can be seen from Fig. 18.5 that 

r* m = a 2 — (a — h m ) 2 = m \) 2 ~ ( b + h m) z 

where a — radius of the wave surface 

r m = radius of the outer boundary of the m-th zone. 

Squaring the terms in parentheses, we get 

r*. = 2 ah m - h* m = bm\ +-m J (^-) 2 - 2 bh m - h* m 


(18.5) 



Difraction of Light 


389 


whence 


, 6mX+m 4 (X/2) # 
hm = — W+b ) — 


( 18 . 6 ) 

Restricting ourselves to a consideration of not too great m 1 s, we may 
disregard the addend containing X 2 owing to the smallness of X. 
In this approximation 

^ m== 2(a+6) v!8.7) 

The area of a spherical segment is S — 2nRh (here R is the radius 
of the sphere and h is the height of the segment). Hence, 


S m = 2 nah m = rnk 


and the area of the m-th zone is 


A S m = i S m — S 


m-i 


nabX 
a + b 


The expression we have obtained does not depend on m. This signi- 
fies that when m is not too great, the areas of the Fresnel zones are 
approximately identical. 

We can find the radii of the zones from Eq. (18.5). When m is not 
too great, the height of a segment h m cty and we can therefore con- 
sider that rfn = 2 ah m . Substituting for h m its value from Eq. (18.7), 
we get the following expression for the radius of the outer boundary 
of the m-th zone: 


r m = ~~b ml 0 8 - 8 ) 

If w*e assume’ that a = b = 1 m and X == 0.5 p,m, then we get a 
value of r x =0.5 mm for the radius of the first (central) zone. The 
radii of the following zones grow as \ m. 

Thus, the areas of the Fresnel zones are approximately the same. 
The distance b m from a zone to point P slowly increases with the 
zone number m. The angle (p between a normal to the zone elements 
and the direction toward point P also grows with m. All this leads 
to the fact that the amplitude A m of the oscillation produced by 
the m-th zone at point P diminishes monotonously with increasing 
m. Even at very high values of m, when the area of a zone begins to 
grow appreciably with m [see Eq. (18.6)], the decrease in the factor 
K (<p) overbalances the increase in A S m , so that A m continues to 
diminish. Thus, the amplitudes of the oscillations produced at 
point P by Fresnel zones form a monotonously diminishing sequence: 

A<i ]> A 3 A m _i 2> A m A m +i ]> . . • 


The phases of the oscillations produced by adjacent zones differ 
by ji. Therefore, the amplitude A of the resultant oscillation at 



390 


Optics 


point P can be represented in the form 

A = A x — A 2 + A 3 — A x + . . . (18.9) 

This expression includes all the amplitudes from odd zones with 
one sign and from even zones with the opposite one. 

Let us write Eq. (18.9) in the form 

A = J Y + (-j" — A 2+-Y-) + ('X~ i4 * + 'r) + * * • ( 18 - 10 ) 

Owing to the monotonous diminishing of A my we can approximately 
assume that 

A Am-i -f- ^m+i 

^m— 2 

The expressions in parentheses will therefore vanish, and Eq. (18.10) 
will be simplified as follows: 

^=4 (is.id 

According to Eq. (18.11), the amplitude produced at a point P 
by an entire spherical wave surface equals half the amplitude produc- 
ed by the central zone alone. If we put in the path of a wave an opaque 
screen having an aperture that leaves only the central Fresnel zone 
open, the amplitude at point P will equal A ly i.e. it will be double 
the amplitude given by Eq. (18.11). Accordingly, the intensity of 
the light at point P will in this case be four times greater than when 
there are no barriers between points S and P. 

Now let us solve the problem on the propagation of light from 
source S to point P by the method of graphical addition of ampli- 
tudes. We shall divide the wave surface into annular zones similar to 
Fresnel zones, but much smaller in width (the path difference from 
the edges of a zone to point P is a small fraction of k the same for 
all zones). We shall depict the oscillation produced at point P by 
each of the zones in the form of a vector whose length equals the 
amplitude of the oscillation, while the angle made by the vector 
with the direction taken as the beginning of measurement gives the 
initial phase of the oscillation (see Sec. 7.7 of Vol. I, p. 203). The 
amplitude of the oscillations produced by such zones at point P 
slowly diminishes from zone to zone. Each following oscillation lags 
behind the preceding one in phase by the same magnitude. Hence, 
the vector diagram obtained when the oscillations produced by the 
separate zones are added has the form shown in Fig. 18.6. 

If the amplitudes produced by the individual zones were the same, 
the tail of the last of the vectors shown in Fig. 18.6 would coincide 
with the tip of the first vector. Actually, the value of the amplitude 
diminishes, although very slightly. Hence, the vectors form a broken 
spiral-shaped line instead of a closed figure. 



Difraction of Light 


391 


In the limit when the widths of the annular zones tend to zero 
(their number will grow unlimitedly), the vector diagram has the 
form of a spiral winding toward point C (Fig. 18.7). The phases 
of the oscillations at points O and 1 differ by ji (the infinitely small 
vectors forming the spiral have opposite directions at these points). 




Fig. 18.6 


Fig. 18.7 


Consequently, part 0-1 of the spiral corresponds to the first Fresnel 
zone. The vector drawn from point O to point 1 (Fig. 18.8a) depicts the 
oscillation produced at point P by this zone. Similarly, the vector 
drawn from point 1 to point 2 (Fig. 18.86) depicts the oscillation 

/ 



{Fig. 18.8 


produced by the second Fresnel zone. The oscillations from the 
first and second zones are in counterphase; accordingly, vectors 01 
and 12 have opposite directions. 

The oscillation produced at point P by the entire wave surface is 
depicted by vector OC (Fig. 18.8c). Inspection of the figure shows 
that the amplitude in this case equals half the amplitude produced 
by the first zone. We have obtained this result algebraically earlier 
(see Eq. (18.11)1. We shall note that the oscillation produced by the 
inner half of the first Fresnel zone is depicted by vector OB (Fig. 18. 8d). 
Thus, the action of the inner half of the first Fresnel zone is not equi- 
valent to half the action of the first zone. Vector OB is ]/ 2 times 
greater than vector OC . Consequently, the intensity of the light pro- 
duced by the inner half of the first Fresnel zone is double the inten- 
sity produced by the entire wave surface. 



392 


Optics 


The oscillations from the even and odd Fresnel zones are in coun- 
terphase and, therefore, mutually weaken one another. If we would 
place in the path of the light wave a plate that would cover all the 

even or odd zones, the intensity of the light at 
point P would sharply grow. Such a plate, 
•known as a zone one, functions like a converging 
lens. Figure 18.9 shows a plate covering the 
even zones. A still greater effect can be achieved 
by changing the phase of the even (or odd) 
zone oscillations by n instead of covering these 
zones. This can be done with the aid of a trans- 
parent plate whose thickness at the places cor- 
responding to the even or odd zones differs by 
Fig. 18.9 a properly selected value. Such a plate is 

called a phase zone plate. In comparison with 
the amplitude zone plate covering zones, a phase plate produces 
an additional two-fold increase in the amplitude, and a four-fold 
increase in the light intensity. 



18.4. Fresnel Diffraction 
from Simple Barriers 

The methods of algebraic and graphical addition of amplitudes 
treated in the preceding section make it possible to solve a number 
of problems involving the diffraction of light. 




Diffraction from a Round Aperture. Let us put an opaque screen 
with a round aperture of radius r 0 cut out in it in the path of a spher- 
ical light wave. We shall arrange the screen so that a perpendicular 
dropped from light source S passes through the centre of the aperture 
(Fig. 18.10). Let us take point P on the continuation of this perpen- 
dicular. At an aperture radius r 0 considerably smaller than the lengths 



Diffraction of Light 


393 


a and b shown in the figure, the length a can be considered equal to 
the distance from source S to the barrier, and the length b, to the 
distance from the barrier to point P. If the distances a and b satisfy 
the relation 

U=V (18.12) 

where m is an integer, then the aperture will leave open exactly m 
first Fresnel zones constructed for point P (see Eq. (18.8)]. Hence, 
the number of open Fresnel zones is determined by the expression 

m =4(T+i) ( ,813 > 

According to Eq* (18.9), the amplitude at point P will be 

A = A i — A i + As — A* -f- . . . dzA m (18.14) 

The amplitude A m is taken with a plus sign if m is odd and with a 
minus sign if m is even. Writing Eq. (18.14) in a form similar to 
Eq. (18.10) and assuming that the expressions in parentheses eaual 
zero, we arrive at the equations 


a . Ai . , A m 
A 2 1 2 

( m is odd) 

a i a 

A 2*2 m 

( m is even) 


The amplitudes from two adjacent zones are virtually the same. 
We may therefore replace {A m ^ x l2) — A m with —A m / 2. The result is 

< 18 - 15 > 

where the plus sign is taken for odd and the minus sign for even rri* s. 

The amplitude A m differs only slightly from A 1 for small m 1 s. 
Hence, with odd m' s, the amplitude at point P will approximately 
equal A ly and at even m’s, zero. It is easy to obtain this result with 
the aid of the vector diagram shown in Fig. 18.7. 

If we remove the barrier, the amplitude at point P will become 
equal to AJ 2 [see Eq. (18.11)]. Thus, a barrier with an aperture 
opening a small odd number of zones not only does not weaken the 
illumination at point P , but, on the contrary, leads to an increase 
in the amplitude almost twice, and of the intensity, almost four 
times. 

Let us determine the nature of the diffraction pattern that will 
be observed on a screen placed after the barrier (see Fig. 18.10). 
Owing to the symmetrical arrangement of the aperture relative to 
straight line SP , the illumination at various points of the screen 
will depend only on the distance r from point P. At this point itself. 


394 


Optic 9 


the intensity will reach a maximum or a minimum depending on 
whether the number of open (effective) Fresnel zones is even or odd. 
Assume, for example, that this number is three. In this case, we 
get a maximum of intensity at the centre of the diffraction pattern. 
A’ pattern of the Fresnel zones for point P is given in Fig. 18.11 a. 



Now let us move along the screen to point P\ The pattern of the 
Fresnel zones for point P' limited by the edges of the aperture has 
the form shown in Fig. 18.116. The edges of the aperture will obstruct 
a part of the third zone, and simultaneously the fourth zone will be 



Odd m Even m 


Fig. 18.12 

partly opened. As a result, the intensity of the light diminishes, 
and reaches a minimum at a certain position of point P\ If we move 
along the screen to point P ”, the edges of the aperture will partly 
obstruct not only the third, but also the second Fresnel zone, simul- 
taneously partly opening the fifth zone (Fig. 18.11c). The result 
will be that the action of the open sections of the odd zones will 
overbalance the action of the open sections of the even zones, and 





Diffraction of Light 


395 


the intensity will reach a maximum. True, this maximum will be 
weaker than that observed at point P . 

Thus, the diffraction pattern produced by a round aperture has 
the form of alternating bright and dark concentric rings. There will 
be either a bright (m is odd) or a dark (m is even) spot at the centre 
of the pattern (Fig. 18.12). The variation in the intensity I with the 
distance r from the centre of the pattern is shown in Fig. 18.106 
(for an Odd m) and in Fig. 18.10c (for an even m ). When the screen 
is moved parallel to itself along straight line SP , the patterns shown 



Fig. 18.13 

in Fig. 18.12 will replace one another [according to Eq. (18.131, 
when 6 changes, the value of m becomes odd and even alternately!. 

If the aperture opens only a part of the central Fresnel zone, a 
blurred bright spot is obtained on the screen; there is no alternation 
of bright and dark rings in this case. If the aperture opens a great 
number of zones, the alternation of the bright and dark rings is 
observed only in a very narrow region on the boundary of the geo- 
metrical shadow; inside this region the illumination is virtually 
constant. 

Diffraction from a Disk. Let us place an opaque disk of radius 
r 0 between light source S and observation point P (Fig. 18.13). 
If the disk covers m first Fresnel zones, the amplitude at point P 
will be 

A = A m+i - A m+i + A m+S - . . . (*^-A m+i + ^p-) + . . . 

The expressions in parentheses can be assumed to equal zero, con- 
sequently 

A = A^± ( 18 . 16 ) 

Let us determine the nature of the pattern obtained on the screen 
(see Fig. 18.13). It is obvious that the illumination can depend only 


396 


Optics 


on the distance r from point P. With a small number of covered 
zones, the amplitude A m + X differs slightly from A x . The intensity 
at point P will therefore be almost the same as in the absence of 
a barrier between source S and point P [see Eq. (18.11)1. For point 
P* displaced relative to point P in any radial direction, the disk 
will cover a part of the (m + l)-th Fresnel zone, and part of the 
m-th zone will be opened simultaneously. This will cause the inten- 
sity to diminish. At a certain position of point P\ the intensity 
will reach its minimum. If the distance from the centre of the pat- 
tern is still greater, the disk will cover 
additionally a part of the (m + 2)-th 
zone, and a part of the (m — l)-th zone 
will be opened simultaneously. As a re- 
sult, the intensity grows and reaches a 
maximum at point P". 

# Thus, the diffraction pattern for an 
opaque disk has the form of alternating 
bright and dark concentric rings. The 
centre of the pattern contains a bright 
spot (Fig. 18.14). The light intensity I var- 
ies with the distance r from point P as 
shown in Fig. 18.136. 

If the disk covers only a small part of 
the central Fresnel zone, it does not form 
a shadow at all — the illumination of the 
screen everywhere is the same as in the absence of barriers. If the disk 
covers many Fresnel zones, alternation of the bright and dark rings 
is observed only in a narrow region on the boundary of the geomet- 
rical shadow. In this case, A m ^ i <C A x , so that the bright spot at 
the centre is absent, and the illumination in the region of the geo- 
metrical shadow equals zero practically everywhere. 

The bright spot at the centre of the shadow formed by a disk 
was the cause of an incident between Simeon Poisson and Augustin 
Fresnel. The Paris Academy of Sciences announced the diffraction 
of light as the topic for its 1818 prize. The organizers of the com- 
petition were advocates of the corpuscular theory of light and were 
sure that the papers submitted to the competition would bring 
a final victory to their theory. Fresnel submitted a paper, however, 
in which all the optical phenomena known at that time were explained 
from the wave viewpoint. In considering this paper, Poisson, 
who was a member of the competition committee, gave attention 
to the fact that an “absurd” conclusion follows from Fresnel’s theory: 
there must be a bright spot at the centre of the shadow* formed by 
a small disk. D. Arago immediately conducted an experiment and 
found that such a spot does indeed exist. This brought victory and 
all-round recognition to the wave theory of light. 



Fig. 18.14 



Diffraction of Light 


397 


Diffraction from the Straight Edge of a Half-Plane. Let us put 
an opaque half-plane with a straight edge in the path of a light 
wave (which we shall consider to be plane for simplicity). We shall 
arrange this half-plane so that it coincides with one of the wave 
surfaces. We shall place a screen parallel to the half-plane at a dis- 
tance b behind it and take point P on the screen (Fig. 18.15). Let 
us divide the open part of the wave surface into zones having the 
form of very narrow straight strips parallel to the edge of the half- 
plane. We shall choose the width of the' zones so that the distances 




Fig. 48.16 


from point P to the edges of any zone measured in the plane of the 
drawing differ by the same amount A. When this condition is obser- 
ved, the oscillations set up at point P by the adjacent zones will 
differ in phase by a constant value. 

We shall assign the numbers 7, 2, 3, etc. to the zones at the right 
of point P, and the numbers l r , 2 ' , 3', etc. to those at the left of this 
point. The zones numbered m and m' have an identical width and 
are symmetrical relative to point P. Therefore, the oscillations pro- 
duced by them at P coincide in amplitude and in phase. 

To establish the dependence of the amplitude on the zone number 
m , let us assess the areas of the zones. A glance at Fig. 18.16 shows 
that the total width of the first m zones is 

d, + d 2 + • . . -M m = l/>+ mA) 2 — b*=V 26mA + m*A 2 

Since the zones are narrow, we have A <C b. Consequently, when m 
is not very great, we may ignore the quadratic term in the radicand. 
This yields 

d\ + d 2 + * • + d m = V 2bm& 

Assuming in this equation that m = 1, we find that d l = ]/ 2&A. 
Hence, we can write the expression for the total width of the first m 
zones as follows 


d\ + d 2 + . . . + d m = \/ln 


398 


Optics 


whence 

d m ^=d i (y m — — l) (18.17) 

Calculations by Eq. (18.17) show that 

d x : d 2 : d 3 : d 4 : . . . = 1 : 0.41 : 0.32 : 0.27 : . . . (18.18) 

The areas of the zones are in the same proportion. 

Examination of Eq. (18.18) shows that the amplitude of the oscil- 
lations set up at point P by the individual zones initially (for the 
first zones) diminishes very rapidly, and then this diminishing 
becomes slower. For this reason, the broken line obtained in the 



Fig. 18.17 Fig. 18.18 


graphical addition of the oscillations produced by the straight zones 
first has a gentler slope than that for annular zones (the areas of 
which in a similar construction are approximately equal). Both 
vector diagrams are compared in Fig. 18.17. In both cases, the lag 
in phase of each following oscillation has been taken the same. The 
value of the amplitude for the annular zones (Fig. 18.17a) has been 
taken constant, and for the straight zones (Fig. 18.176), diminishing 
in accordance with proportion (18.18). The graphs in Fig. 18.17 are 
approximate. In an exact construction of these graphs, account must 
be taken of the dependence of the amplitude on r and qp [see Eq. 
(18.1)1. This does not affect the general nature of the diagrams, 
however. 

Figure 18.176 shows only the oscillations produced by the zones 
to the right of point P . The zones numbered m and m' are symmetr- 
ical relative to P . It is therefore natural, when constructing the 
diagram, to arrange the vectors depicting the oscillations corres- 
ponding to these zones symmetrically relative to the origin of coor- 
dinates O (Fig. 18.18). If the width of the zones is made to tend 
to zero, the broken line shown in Fig. 18.18 will transform into 
a smooth curve (Fig. 18.19) called a Cornu spiral* 



Difraction of Light 


399 


The equation of a Cornu spiral in the parametric form is 

|= j cos ^Y~du, q= ( sin-^p-du (18.19) 

These integrals are known as Fresnel integrals. They can be solved 
only numerically, and tables are available that can be used to 



find the values of the integrals for various t>’s. The meaning of the 
parameter v is that | v | gives the length of the arc of the Cornu spiral 
measured from the origin of coordinates. 

The figures along the curve in Fig. 18.19 give the values of the 
parameter v. Points F x and F 2 which are asymptotically approached 
by the curve when v tends to + co and — ooare called the focal points 
or poles of the Cornu spiral. Their coordinates are 

1= +Y ’ T 1 = +T for point F t 

£=-i., r|= — for point F 2 




400 


Optics 


The right-hand curl of the spiral (section OF x ) corresponds to zones 
to the right of point P> and the left-hand curl (section OF 2 ) to zones 
to the left of this point. 

Let us find the derivative dx\!d\ for the point of the curve corres- 
ponding to a given value of the parameter u. According to Eq. (18.19), 
the values 


= cos — y- dv, dtj = sin — y du 

correspond to the increment dv of u . Consequently, dx\!d\ = 
= tan (nv*/2). At the same time, drj/d£ = tan 0, where 0 is the 
angle of inclination of a tangent to the curve at the given point. 
Thus, 

0 = -£-i> 2 (18.20) 

It thus follows that at the point corresponding to v = 1 a tangent 
to the Cornu spiral is perpendicular to the 1-axis. When v = 2, 
the angle 0 is 2it, so that a tangent is parallel to the £-axis. When 
v = 3, the angle 0 is 9 ji/ 2, so that a tangent is again perpendicular 
to the £-axis, and so on. 

The Cornu spiral makes it possible to find the amplitude of a light 
oscillation at any point on a screen. We shall characterize the posi- 
tion of the point by the coordinate x measured from the boundary of 
the geometrical shadow (see Fig. 18.15). All the hatched zones will 
be covered for point P on the boundary of the geometrical shadow 
(x =0). The right-hand curl of the spiral corresponds to oscillations 
from the unhatched zones. Hence, the resultant oscillation will be 
depicted by a vector whose origin is at point O and whose end is at 
point F x (Fig. 18.20a). When point P is displaced to the region of 
the geometrical shadow, the half-plane covers a greater and greater 
number of unhatched zones. Therefore, the beginning of the resul- 
tant vector moves along the right-hand curl in the direction of 
pole F x (Fig. 18.206). As a result, the amplitude of the oscillation 
monotonously tends to zero. 

If point P is displaced from the boundary of the geometrical shad- 
ow to the right, in addition to the unhatched zones a constantly 
growing number of hatched ones will be uncovered. Therefore, the 
tip of the resultant vector slides along the left-hand curl of the spiral 
in a direction to pole F z . The amplitude passes through a number 
of maxima (the first of them equals the length of segment MF X 
in Fig. 18.20c) and minima (the first of them equals the length of 
segment NF X in Fig. 18.20d). When the wave surface is completely 
uncovered, the amplitude equals the length of F 2 F X (Fig. 18.20c), 
i.e. is exactly double the amplitude on the boundary of the geomet- 
rical shadow (see Fig. 18.20a). Accordingly, the intensity on the 



Diffraction of Light 


401 


boundary of the geometrical shadow is one-fourth of the intensity J 0 
obtained on the screen in the absence of barriers. 

The dependence of light intensity I on the coordinate x is shown 
in Fig. 18.21. Inspection of the figure shows that upon a transition 






to the region of the geometrical shadow, the intensity gradually 
tends to zero instead of changing in'a jump. A number of alternating 
maxima and minima of the intensity are to the right of the boundary 
of the geometrical shadow. Calculations show that at b = 1 m 
and X — 0.5 p,m the coordinates of the maxima (see Fig. 18.21) 
have the following values: —0.61 mm, x 2 =1.17 mm, x s = 

= 1.54 mm, x k = 1.85 mm, etc. With a change in the distance b 
from the half-plane to the screen, the values of the coordinates of 
the maxima and minima vary as Y b . It can be seen from the above 


402 


Optics 


data that the maxima are quite dense. The Cornu curve can also 
be used to find the relative value of the intensity at the maxima and 
minima. We get the value of 1.37 I 0 for the first maximum and 
0.78/ 0 for the first minimum. 



Fig. 18.22 


Wave surface 



i 

i 

» 


p” p* p 


Fig. 18.23 


Figure 18.22 contains a photograph of the diffraction pattern 
produced by the edge of a half-plane. 

Diffraction from a Slit. An infinitely long slit can be formed by 
placing two half-planes facing opposite directions next to each 

other. Therefore, the problem on the 
Fresnel diffraction from a slit can be 
solved with the aid of a Cornu spiral. 
We shall consider that the wave sur- 
face of the incident light, the plane of 
the slit, and the screen on which a dif- 
fraction pattern is observed are parallel 
to one another (Fig. 18.23). 

For point P opposite the middle of 
the slit, the tip and the tail of the re- 
sultant vector are at points on the spi- 
ral that are symmetrical relative to the 
origin of coordinates (Fig. 18.24). If we 
move to point P' opposite an edge of 
the slit, the tip of the resultant vec- 
tor will move to the middle of the spiral O . The tail of the vector 
will move along the spiral in the direction of pole F v Upon motion 
into the region of the geometrical shadow, the tip and the tail of 
the resultant vector will slide along the spiral and in the long run 
will be at the smallest distance apart (see the vector in Fig. 18.24 
corresponding to point P "). The intensity of the light reaches a 
minimum here. Upon further sliding along the spiral, the tip and 




Diffraction of Light 


403 


tail of the vector will move apart again, and the intensity will grow. 
The same will occur when we move from point P in the opposite 
direction because the diffraction pattern is symmetrical relative 
to the middle of the slit. 

If we change the width of the slit by moving the half-planes in 
opposite directions, the intensity at middle point P will pulsate. 




Fig. 18.25 

alternately passing through maxima (Fig. 18.25a) and minima that 
differ from zero (Fig. 18.256). 

Thus, a Fresnel diffraction pattern from a slit is either a bright 
(for the case shown in Fig. 18.25a) or a relatively dark (for the case 
shown in Fig. 18.256) central fringe at both sides of which there are 
alternating dark and bright fringes symmetrical relative to it. 

With a great width of the slit, the tip and tail of the resultant 
vector for point P are on the internal turns of the spiral near poles 
F x and F 2 . Therefore, the intensity of the light at the points opposite 
the slit will be virtually constant. A system of closely spaced narrow 
bright and dark fringes is formed only on the boundaries of the geo- 
metrical shadow. 

We must note that all the results obtained in the present section 
hold provided that the coherence radius of the light wave falling 
on the barrier greatly exceeds the characteristic dimension of 
the barrier (the diameter of the aperture or disk, the width of 
the slit, etc.). 

18.5. Fraunhofer Diffraction from a Slit 

Assume that a plane light wave falls on an infinitely long* slit 
(Fig. 18.26). Let us place a converging lens behind the slit and 
a screen in the focal plane of the lens. The wave surface of the inci- 

* In practice, it is sufficient that the length of the slit be many times its 
width. 


404 


Optics 


dent wave, the plane of the slit, and the screen are parallel to one 
another. Since the slit is infinite, the pattern observed in any plane 
at right angles to the slit will be the same. It is therefore sufficient 
to investigate the nature of the pattern in one such plane, for example, 
in that of Fig. 18.26. All the quantities introduced in the following, 
in particular the angle qp made by a ray with the optical axis of the 
lens, relate to this plane. 

Let us divide the open part of the wave surface into elementary 
zones of width dx parallel to the edges of the slit. The secondary 

waves emitted by the zones in the 
direction determined by the angle 
qp will gather at point P of the screen. 
Each elementary zone will pro- 
duce the oscillation dE at point P. 
The lens will gather plane (and not 
spherical) waves in the focal plane. 
Therefore, the factor 1/r in Eq. (18.1) 
for dE will be absent for Fraunho- 
fer diffraction. Limiting ourselves 
to a consideration of not too great 
angles qp, we can assume that the 
coefficient K in Eq. (18.1) is con- 
stant. Hence, the amplitude of the 
oscillation produced by a zone at any point of the screen will depend 
only on the area of the zone. The area is proportional to the width 
dx of a zone. Consequently, the amplitude dA of the oscillation dE 
produced by a zone of width dx at any point of the screen will have 
the form 

dA = C dx 

where C is a constant. 

Let A 0 stand for the algebraic sum of the amplitudes of the oscil- 
lations produced by all the zones at a point of the screen. We can 
find A 0 by integrating dA over the entire width of the slit b: 

+ 6/2 

A 0 = j dA = J Cdx = Cb 

-b / 2 

Hence, C = Ajb and, therefore, 

dA = -^-dx 

0 

Now let us find the phase relations between the oscillations dE . 
We shall compare the phases of the oscillations produced at point P 
by the elementary zones having the coordinates O and x (Fig. 18.26)* 
The optical paths OP and QP are tautochronous (see Fig. 16.20). 
Therefore, the phase difference between the oscillations being con- 




Diffraction of Light 


405 


sidered is formed on the path A equal to x sin cp. If the initial phase 
of the oscillation produced at point P by the elementary zone at the 
middle of the slit (q> = 0) is assumed to equal zero, then the initial 
phase of the oscillation produced by the zone with the coordinate x . 
will be 

o A 2n 

— 2n = x sin <p 

where X is the wavelength in the given medium. 

Thus, the oscillation produced by the elementary zone with the 
coordinate x at point P (whose position is determined by the angle (p) 
can be written in the form 

(-^-dx^exp [* ( x sin <P ) J (18.21) 

(we have in mind the real part of this expression). 

Integrating Eq. (18.21) over the entire width of the slit, we shall 
find the resultant oscillation produced at point P by the part of 
the wave surface uncovered by the slit: 

+6/2 

£$= j -^-exp^i^cof — -^-xsin<pj J dx 
- 6/2 

Let us put the multipliers not depending on x outside the integral 
In addition, we shall introduce the symbol 


y = -~sin <p 


As a result, we get 


— 6/2 * 1 

- {-$- Trio I****- - **" {-# ir i* w - 

The expression in braces determines the complex amplitude 
of the resultant oscillation. Taking into account that the difference 
between the exponents divided by 2i is sin yb (see Sec. 7.3 of Vol. I, 
p. 190 et seq.), we can write 


2 a sin y b A sin (ji b sin q>/X) 

Q 0 yb 0 (nbsinq>/X) 


(18.23) 


[we have introduced the value of y from Eq. (18.22)]. 

Equation (18.23) is a real one. Its magnitude is the usual ampli- 
tude of the resultant oscillation: 


A a sin ( nb sin <p/X) 

^ 0 («6 sin <p/X) 


(18.24) 



406 


Optics 


For a point opposite the centre of the lens, cp = 0. Introduction 
of this value into Eq. (18.24) gives the value A 0 for the amplitude*. 
This result can be obtained in a simpler way. When cp = 0, the 
oscillations from all the elementary zones arrive at point P in the 
same phase. Therefore, the amplitude of the resultant oscillation 
equals the algebraic sum of the amplitudes of the oscillations being 
added. 

At values of <p satisfying the condition nb sin <pA, = dbkn y i.e. 
when 

b sin <p = ±k\ (k = 1, 2, 3, . . .) (18.25) 

the amplitude vanishes. Thus, condition (18.25) determines 



Fig. 18.27 Fig. 18.28 


the positions of the minima of intensity. We must note that b sin <p 
is the path difference A of the rays travelling to point P from the 
edges of the slit (see Fig. 18.26). 

It is easy to obtain condition (18.25) from the following consider- 
ations. If the path difference A from the edges of the slit is ±k\ y 
the uncovered part of the wave surface can be divided into 2k zones 
equal in width, and the path difference from the edges of each zone 
will be X/2 (see Fig. 18.27 for k = 2). The oscillations from each 
pair of adjacent zones mutually destroy each other, so that the resul- 
tant amplitude vanishes. If the path difference A for point P is 

± k, the number of zones will be odd, the action of one of 

them will not be compensated, and a maximum of intensity is 
observed. 

The intensity of light is proportional to the square of the ampli- 
tude. Hence, in accordance with Eq. (18.24), 


j j sin 2 (ji b sin <p/X) 

(ji 6 sin <pA) 2 


(18.26) 


* We remind our reader that lim (sin uiu) = 1 (at small values of u we may 
assume that sin u « u). 



Diffraction of Light 


407 


where I 0 is the intensity at the middle of the diffraction pattern 
(opposite the centre of the lens), and I <p is the intensity at the point 
whose position is determined by the given value of ip. 

We find from Eq. (18.26) that / _<p = / v . This signifies that the 
diffraction pattern is symmetrical relative to the centre of the lens. 
We must note that when the slit is displaced parallel to the screen 
(along the x-axis in Fig. 18.26), the diffraction pattern observed on 
the screen remains stationary (its middle is opposite the centre of 
the lens). Conversely, displacement of the lens with the slit sta- 
tionary is attended by the same displacement of the pattern on the 
screen. 

A graph of function (18.26) is depicted in Fig. 18.28. The values 
of sin q> are laid off along the axis of abscissas, and the intensity / 9 
along the axis of ordinates. The number of intensity minima is 
determined by the ratio of the width of a slit b to the wavelength X. 

It can be seen from condition (18.25) that sin <p = dtkX/b. The mag- 
nitude of sin <p cannot exceed unity. Hence, kX/b^. 1, whence 

(18.27) 

At a slit width less than a wavelength, minima do not appear at all. 
In this case, the intensity of the light monotonously diminishes 
from the middle of the pattern toward its edges. 

The values of the angle <p obtained from the condition b sin q> = 
= ±X correspond to the edges of the central maximum. These 
values are ±arcsin (X/t). Consequently, the angular width of the 
central maximum is 

6cp 2 arcsin y (18.28) 

When b » X, the value of sin (X/fr) can be assumed equal to X/6. 
The equation for the angular width of the central maximum is thus 
simplified as follows: 

= (18.29) 

Let us solve the problem on the Fraunhofer diffraction from a slit . 
by the method of graphical summation of the amplitudes. We divide 
the open part of the wave surface into very narrow zones of an iden- 
tical width. The oscillation produced by each of these zones has 
the same amplitude AA and lags in phase behind the preceding 
oscillation by the same value 6 that depends on the angle <p deter- 
mining the direction to the point of observation P. When <p = 0, 
the phase difference 6 vanishes, and the vector diagram has the form 
shown in Fig. 18.29a. The amplitude of the resultant oscillation A 0 



408 


Optics 


equals the algebraic sum of the amplitudes of the oscillations being 
added. When A = b sin 9 = X/2, the oscillations from edges of 



Fig. 18.29 


sity of the first maximum is I x 
find the relative intensity of the 
As a result, we get the following 


the slit are in counterphase. Ac- 
cordingly, the vectors A A arrange 
themselves along a semicircle of 
length A 0 (Fig. 18.296). Hence, 
the resultant amplitude is 2AJn. 
When A = 6 sin cp = X, the os- 
cillations from the edges of the 
slit differ in phase by 2ji. The 
corresponding vector diagram is 
shown in Fig. 18.29c. The vec- 
tors AA arrange themselves along 
a circle of length A 0 . The resul- 
tant amplitude is zero — the first 
minimum is obtained. The first 
maximum is obtained at A = 
= b sin cp = 3X/2. In this case, 
the oscillations from the edges 
of the slit differ in phase by 3jx. 
Constructing sequentially the 
vectors AA, we travel one-and-a- 
half times around a circle of dia- 
meter A x — 2AjZn (Fig. 18.29d). 
It is exactly the diameter of this 
circle that is the amplitude of the 
first maximum. Thus, the inten- 
= (2/3n) 2 Iq » 0.045/q. We can 
other maxima in a similar way. 
proportion: 


Iq : 1 1 : /j : 1$ • . 



= 1 : 0.045 : 0.016 : 0.008 : . . . (18.30) 


Thus, the central maximum considerably exceeds the remaining 
maxima in intensity; the main fraction of the light flux passing 
through the slit is concentrated in it. 

When the width of the slit is very small in comparison with the 
distance from it to the screen, the rays travelling to point P from 
the edges of the slit will be virtually parallel even in the absence 
of a lens between the slit and the screen. Consequently, when a plane 
wave falls on a slit, Fraunhofer diffraction will be observed. All 
the equations obtained above will hold; by cp in them one should 
understand the angle between the direction from any edge of the 
slit to point P and a normal to the plane of the slit. 


Diffraction of Light 


409 


Let us establish a quantitative criterion permitting us to deter- 
mine the kind of diffraction that will occur in each particular case. 
We shall find the path difference of the rays from the edges of the 
slit to point P (Fig. 18.30). We apply the 
cosine law to the triangle with the legs 
r, r + A, and b : 

(r -f- A) 2 = r 2 + & 2 — 2rb cos + 

Simple transformations yield 

2rA + A* = b 2 -f 2 rb sin <p (18.31) 

We are interested in the case when the 
rays travelling from the edges of the slit 
to point P are almost parallel. When this 
condition is observed, A 2 C rA, and we 
can therefore ignore the addend A 1 in 
Eq. (18.31). In this approximation 

A = -|p- + &sin <p (18.32) 

In the limit at r — * oo, we get a value of the path difference A*, = 
= b sin <p that coincides with the expression in Eq. (18.25). 

At finite r’s, the nature of the diffraction pattern will be deter- 
mined by the relation between the difference A — Ac and the wave- 
length X. If 

A — A* (18.33) 

the diffraction pattern will be virtually the same as in Fraunhofer 
diffraction. At A — A« comparable with X (i.e. A — A«> ~ X), 
Fresnel diffraction will take place. 

It follows from Eq. (18.32) that 



A Ago 



6 * 

l 


(here l is the distance from the slit to the screen). Introduction of 
this expression into inequality (18.33) gives the condition ( b 2 /l ) <C 
«X or 


a 




(18.34) 


Thus, the nature oi diffraction depends on the value of the dimen- 
sionless parameter 

(18.35) 


If this parameter is much smaller than unity, Fraunhofer diffraction 
is observed, if it is of the order of unity, Fresnel diffraction is observed. 



410 


Optics 


and, finally, if this parameter is much greater than unity, the 
approximation of geometrical optics is applicable. For convenience 
of comparison, let us write what has been said above in the follow- 
ing form: 

< 1 — Fraunhofer diffraction 
~1 — Fresnel diffraction (18.36) 

1 — geometrical optics 

Parameter (18.35) can be given a visual interpretation. Let us 
take point P opposite the middle of a slit (Fig. 18.31). For this 

point, the number m of Fresnel zones opened 
by the slit is determined by the expression 

( i + / ra y ) 2 = Z2 +( t ) 2 

Opening the parentheses and discarding the 
addend proportional to A, 2 , we get* 

“-W-TT < 1837 ) 

Thus, parameter (18.35) is directly associated 
with the number of uncovered Fresnel zones 
(for a point opposite the middle of the slit). 

If a slit opens a small fraction of the central Fresnel zone (m 1), 
Fraunhofer diffraction is observed. The distribution of the intensity 
in this case is shown by the curve depicted in Fig. 18.28. If a slit 
uncovers a small number of Fresnel zones ( m ~ 1), an image of the 
slit surrounded along its edges by clearly visible bright and dark 
fringes will be obtained on the screen. Finally, when a slit opens 
a large number of Fresnel zones (m 1), a uniformly illuminated 
image of the slit is obtained on the screen. Only at the boundaries 
of the geometrical shadow are there very narrow alternating brighter 
and darker fringes virtually indistinguishable by the eye. 

Let us see how the pattern changes when the screen is moved away 
from the slit. When the screen is near the slit (m » 1), the image 
corresponds to the laws of geometrical optics. Upon increasing the 
distance, we first obtain a Fresnel diffraction pattern which then 
transforms into a Fraunhofer pattern. The same sequence of changes 
is observed when we reduce the width of the slit b without changing 
the distance l. 

It is clear from what has been said above that the value of the 
parameter (18.35) is the criterion of the applicability of geometrical 


* We must note that the number of open zones will be larger for points 
greatly displaced to the region of the geometrical shadow* 



Fig. 18.31 




Diffraction of Light 


411 


optics (it must be much greater than unity) instead of the smallness 
of the wavelength in comparison with the characteristic dimension 
of the barrier (for example, the width of the slit). Assume, for in- 
stance, that both ratios b/’K and I/b equal 100. In this case, )i<6, 
but b 2 /lk = 1, and, therefore, a distinctly expressed Fresnel diffrac- 
tion pattern will be observed. 


18.6. Diffraction Grating 


A diffraction grating is a collection of a large number of identical 
equispaced slits (Fig. 18.32). The distance d between the centres 
of adjacent slits is called the period of the grating. 

Let us place a converging lens parallel to a grating and put a 
screen in the focal plane of the lens. We shall determine the nature 
of the diffraction pattern obtained on 
the screen when a plane light wave falls 
on the grating (we shall consider for 
simplicity’s sake that the wave falls 
normally on the grating). Each slit pro- 
duces a pattern on the screen that is 
described by the curve depicted in 
Fig. 18.28. The patterns from all the 
slits will be at the same place on the 
screen (regardless of the position of the 
slit, the central maximum is opposite 
the centre of the lens). If the oscilla- 
tions arriving at point P from different slits were incoherent, the 
resultant pattern produced by N slits would differ from the pattern 
produced by a single slit only in that all the intensities would grow 
N times. The oscillations from different slits are coherent to a greater 
or smaller extent, however. The resultant intensity will therefore 
differ from NI v II v is the intensity produced by one slit; see Eq. 
(18.26)]. 

We shall assume in the following that the coherence radius of the 
incident wave is much greater than the length of the grating so tfrat 
the oscillations from all the slits can be considered coherent relative 
to one another. In this case, the resultant oscillation at point P 
whose position is determined by the angle q> is the sum of N oscil- 
lations having the same amplitude A ^ shifted relative to one another 
in phase by the same amount 6. According to Eq. (17.47), the inten- 
sity in these conditions is 



Fig. 18.32 


r ___ r sin 2 (/V6/2) 
sin 2 (6/2) 


(18.38) 


(here I m plays the part of / 0 ). 



412 


Optics 


A glance at Fig. 18.32 shows that the path difference from adjacent 
slits is A = d sin qp. Hence, the phase difference is 


. A 2n j . 

► = 2ji = -t— d sin <p 


(18.39) 


where X is the wavelength in the given medium. 

Introducing into Eq. (18.38) Eqs. (18.26) and (18.39) for and 
6, respectively, we get 


j r sin* (it b sin cp/X) sin* (Nnd sin cp/X) 

* gr * o /_ i_ _ « v« _ - o / i _ ? . »<i \ " 


(nft sin cp/X)* sin* (nd sin cp/X) 


(18.40) 


(/ 0 is the intensity produced by one slit opposite the centre of the 
lens). 

The first multiplier of I 0 in Eq. (18.40) vanishes for points for 
which condition (18.25) is observed, i.e., 

b sin qp = ±kk (k = 1, 2, 3, . . .) 


At these points, the intensity produced by each slit individually 
equals zero. 

The second multiplier of I 0 in Eq. (18.40) acquires the value N % 
for points satisfying the condition 

d sin cp = dtmX (m = 0, 1, 2, . . .) (18.41) 

[see Eq. (17.49)1. For the directions determined by this condition, 
the oscillations from individual slits mutually amplify one another. 
As a result, the amplitude of the oscillations at the corresponding 
point of the screen is 

^max ~ Ai4qj (18.42) 

(A<p is the amplitude of the oscillation emitted by one slit at the 
angle qp). 

Condition (18.41) determines the positions of the intensity maxi- 
ma called the principal ones. The number m gives the order of the 
principal maximum. There is only one zero-order maximum, and 
there are two each of the maxima of the 1st, 2nd, etc. orders. 

Squaring Eq. (18.42), we find that the intensity of the principal 
maxima / raax is N 2 times greater than the intensity / * produced 
in the direction qp by a single slit: 

/max = ^ (18.43) 

Apart from the minima determined by condition (18.25), there 
are N — 1 additional minima in each interval between adjacent 
principal maxima. These minima appear in the directions for which 
the oscillations from individual slits mutually destroy one another. 
In accordance with Eq. (17.50), the directions of the additional 



Diffraction of Light 


413 


minima are determined by the condition 

d sin <p = dt -jp- X (18.44) 

(*' = 1,2, .... 1, N + i, .... 2N-i, 2W + 1, ...) 

In Eq. (18.44), k' takes on all integral values except for 0, N , 2 N, . . . 
. . i.e. except for those at which Eq. (18.44) transforms into 
Eq. (18.41). 

It is easy to obtain condition (18.44) by the method of graphical 
addition of oscillations. The oscillations from the individual slits 



Fig. 18.33 

are depicted by vectors of the same length. According to Eq. (18.44), 
each of the following vectors is turned relative to the preceding 
one by the same angle 

o = -y- d sin cp = k 

Therefore when k 9 is not an integral multiple of N, we put the tip 
of the following vector against the tail of the preceding one and 
obtain a closed broken line that completes k ' (when k' < N/2) 
or N — k ' (when k' >N/2) revolutions before the tail of the A’-th 
vector contacts the tip of the first one. The resultant amplitude 
accordingly equals zero. The above is explained in Fig. 18.33 that 
shows the sum of the vectors for N = 9 and for -the values of k 9 
equal to 1, 2, and N — 1 =8. 

Between the additional minima, there are weak secondary maxi- 
ma. The number of such maxima falling to an interval between 
adjacent principal maxima is TV — 2. We showed in Sec. 17.6 that 
the intensity of the secondary maxima does not exceed l/22nd 
of that of the closest principal maximum. 

Figure 18.34 shows a graph of function (18.40) for N — 4 and 
d/b == 3. The dash curve passing through the peaks of the principal 
maxima shows the intensity produced by one slit multiplied by N* 
[see Eq. (18.43)]. At the ratio of the grating period to the slit width 
used in the figure (d/b = 3), the principal maxima of the third, 


414 


Optics 


sixth, etc. orders fall to the minima of intensity from one slit, 
owing to which these maxima vanish. In general, it can be seen 
from Eqs. (18.25) and (18.41) that the principal maximum of the 
m-th order falls to the k - th minimum from one slit if the equation 
mid = k/b or mlk = d/b is satisfied. This is possible if d/b equals 
the ratio of two integers r and s (the case when these integers are 
not great is of practical interest). Here, the principal maximum of the 



Fig. 18.34 


r-th order will be superposed on the s - th minimum from one slit, 
the maximum of the 2 r-th order will be superposed on the 2 s-th 
minimum, etc. As a result, the maxima of orders r, 2r, 3r, etc. will 
be absent. 

The number of principal maxima observed is determined by the 
ratio of the period of the grating d to the wavelength X. The mag- 
nitude of sin <p cannot exceed unity. It therefore follows from Eq. 
(18.41) that 

m<-£ (18.45) 

Let us determine the angular width of the central (zero) maximum. 
The position of the additional minima closest to it is determined 
by the condition d sin <p = ±X/N [see Eq. (18.44)]. Hence, values 
of <p equal to ±arcsin (X/Nd) correspond to these minima. We thus 
obtain the following expression for the angular width of the central 
maximum: 


6 q > 0 = 2 arcsin ~ (18.46) 

(we have taken advantage of the circumstance that X/Nd < 1). 

The position of the additional minima closest to the principal 
maximum of the m-th order is determined by the condition d sin 9 = 
= (m ± 1 IN) X. Hence, for the angular width of the m-th maximum. 


Diffraction of Light 


415 


we get the expression 

6<Pm = arcsin ( m + -jf ) -j — arcsin (m — 

Introducing the notation mX/d = x and X/iVd = Ax, we can write 
this equation in the form 

6<p m = arcsin (x + Ax) — arcsin (x — Ax) (18.47) 

With a great number of slits, the value of Ax = XINd will be very 
small. We can therefore assume that arcsin (x zh Ax) ^ arcsin x ± 



v r 


0 

1 + 2 

□ 


+S 

I 

v r 

v r 


V 


Fig. 18.35 


± (arcsin x)' Ax. The introduction of these values into Eq. (18.47) 
leads to the approximate expression 

6 (p m ~ 2 (arcsin x)' Ax — ==- ■ 1 (18.48) 

When m = 0, this expression transforms into Eq. (18.46). 

The product Nd gives the length of the diffraction grating. Conse- 
quently, the angular width of the principal maxima is inversely 
proportional to the length of the grating. The width 6<p m grows with 
an increase in the order m of a maximum. 

The position of the principal maxima depends on the wave- 
length X. Therefore, when white light is passed through a grating, 
all the maxima except for the central one will expand into a spectrum 
whose violet end faces the centre of the diffraction pattern, and 
whose red end faces outward. Thus, a diffraction grating is a spectral 
instrument. We must note that whereas a glass prism deflects violet 
rays the greatest, a diffraction grating, on the contrary, deflects 
red rays to a greater extent. 

Figure 18.35 shows schematically the spectra of different orders 
produced by a grating when white light is passed through it. At 
the centre is a narrow zero-order maximum; only its edges are coloured 
[according to expression (18.46), 6cp 0 depends on XI. At both 
sides of the central maximum are two first-order spectra, then two 
second-order spectra, etc. The positions of the red end of the m-th 
order spectrum and the violet end of the (m l)-th order one are de- 
termined by the relations 

sin <pt = m -j— » sin <p v = (m+1) d - 




416 


Optics 


where d has been taken in micrometres. When the condition is 
observed that 


0.76m >0.40 (m + 1) 


the spectra of the m-th and (m + l)-th orders partly overlap. The 
inequality gives m > 10/9. Hence, partial overlapping begins from 
the spectra of the second and third orders (see Fig. 18.35, in which 
for illustration the spectra of different orders are displaced relative 
to one another vertically). 

The main characteristics of a spectral instrument are its dispersion 
and resolving power. The dispersion determines the angular or 
linear distance between two spectral lines differing in wavelength 
by one unit (for example by 1 A). The resolving power determines 
the minimum difference between wavelengths 6X at which the two 
lines corresponding to them are perceived separately in the spectrum. 

The angular dispersion is defined as the quantity 



(18.49) 


where 6cp is the angular distance between spectral lines differing 
in wavelength by 6X. 

To find the angular dispersion of a diffraction grating, let us 
differentiate condition (18.41) for the principal maximum at the 
left with respect to q> and at the right with respect to X. Omitting 
the minus sign, we get 


whence 


(d cos q>) 6xp = m 6X 



m 

d cos cp 


(18.50) 


Within the range of small angles*, cos <p « 1. We can therefore 
assume that 


D 



(18.51) 


It can be seen from expression (18.51) that the angular dispersion 
is inversely proportional to the grating period d. The higher the 
order m of a spectrum, the greater is the dispersion. 

Linear dispersion is defined as the quantity 

Dm—jfc (18.52) 


where 61 is the linear distance on a screen or photographic plate 
between spectral lines differing in wavelength by 6X. A glance at 
Fig. 18.36 shows that for small values of the angle <p we can assume 
that 61 /'6<p, where /' is the focal length of the lens gathering 


Diffraction of Light 


417 


the diffracted rays on a screen. Consequently, the linear dispersion 
is associated with the angular dispersion D by the relation 

Dnn-f'D 


Taking expression (18.51) into consideration, we get the following 
equation for the linear dispersion of a diffraction grating (with 
small <p’s): 


A»-f4 


(18.53) 


The resolving power of a spectral instrument is defined as the 
dimensionless quantity 




X 

6X 


(18.54) 


where 6% is the minimum difference between the wavelengths of two 
spectral lines at which these lines are perceived separately. 




The possibility of resolving (i.e. perceiving separately) two close 
spectral lines depends not only on the distance between them (that 
is determined by the dispersion of the instrument), but also on the 
width of the spectral maximum. Figure 18.37 shows the resultant 
intensity (solid curves) observed in the superposition of two close 
maxima (the dash curves). In case a, both maxima are perceived as a 
single one. In case 6, there is a minimum between the maxima. Two 
close maxima are perceived by the eye separately if the intensity 
in the interval between them is not over 80 per cent of the intensity 
of a maximum. According to the criterion proposed by the British 
physicist John Rayleigh (1842-1919), such a ratio of the intensities 
occurs if the middle of one maximum coincides with the edge of 
another one (Fig. 18.376). Such a mutual arrangement of the maxima 
is obtained at a definite (for the given instrument) value of 6X. 

Let us find the resolving power of a diffraction grating. The posi- 
tion of the middle of the m-th maximum for the wavelength X + 6X 



418 


Optics 


is determined by the condition 

(t Sin (pma^ — TTt (X -f- 

The edges of the m-th maximum for the wavelength X are at angles 
complying with the condition 

d sin cp ra ,„ = (m 

The middle of the maximum for the wavelength k + 6k coincides 
with the edge of the maximum for the wavelength k if 

m (k + &k) =: ( m + -ft ) ^ 

whence 

™6X = -^- 

Solving this equation relative to X/6X, we get an expression for the 
resolving power: 

R = rnN (18.55) 

Thus, the resolving power of a diffraction grating is proportional 
to the order m of the spectrum and the number of slits N . 

Figure 18.38 compares the diffraction patterns obtained for two 
spectral lines with the aid of gratings differing in the values of the 

dispersion D and the resolving power /?. 
Gratings I and II have the same resolving 
power (they have the same number of slits 
N), but a different dispersion (in grating /, 
the period d is double and the dispersion 
D is half of the respective quantities of 
grating II). Gratings II and III have the 
same dispersion (they have the same d’s), 
but a different resolving power (the number 
of slits N and the resolving power R of grat- 
ing II are double the respective quantities 
of grating III). 

Transmission and reflecting diffraction 
Fig. 18.38 gratings are in use. Transmission gratings 

are made from glass or quartz plates on 
whose surface a special machine using a diamond cutter makes a num- 
ber of parallel lines. The spaces between these lines are the slits. 

Reflecting gratings are applied with the aid of a diamond cutter 
on the surface of a metal mirror. Light falls on a reflecting grating 
at an acute angle. A grating of period d functions in the same way 
as a transmission grating with the period dcos 0, where 0 is the 
angle of incidence of the light, would function with the light falling 




Diffraction of Light 


419 


normally. This makes it possible to observe a spectrum when light 
is reflected, for example, from a gramophone record having only 
a few lines (grooves) per millimetre if it is placed so that the angle 
of incidence is close to ji/2. The American physicist Henry Rowland 
(1848-1901) invented a concave reflecting grating which focuses 
the diffraction spectra by itself (without a lens). 

The best gratings have up to 1200 lines per mm (d zz 0.8 pm). 
It can be seen from Eq. (18.45) that no second-order spectra are 
observed in visible light with such a period. The total number of 
lines in such gratings reaches 200 000 (they are about 200 mm long). 
With a focal length of the instrument /' = 2 m, the length of the 
visible first-order spectrum in this case is over 700 mm. 


18.7. Diffraction of X-Rays 

Let us place two diffraction gratings one after the other so that 
their lines are mutually perpendicular. The first grating (whose 
lines, say, are vertical) will produce a number of maxima in the 
horizontal direction. Their positions are determined by the condition 

d x sin <p* = {m 1 =0, 1 , 2, . . .) (18.56) 

The second grating (with horizontal lines) will divide each of the 


- 2:2 

0 

1 

- 7:2 

i 

0:2 

• 

7:2 

? 

2:2 

o 

I 

1 

- 2 :J 

0 

1 

» 

- 7 ;T 

i 

-k- 

t 

1 

7/7 

i 

I 

zr 

o 

i 

1 

- 2:0 

0 

1 

i 

-w 

— i 


i 

7:0 

~ i 

2:0 

o 

1 

-J:-r 

0 

1 

- 7,-7 

i 

• ** 

V -7 

i 

2:-7 

o 

1 

- 2:-2 

i 

- 7,-2 

i 

0-2 

i 

7;-2 

1 

2:-2 


O— — — O— — o— — — — o — o 


Fig. 18.39 



beams formed in this way into vertically arranged maxima whose 
positions are determined by the condition 

d 2 s in (p 2 = ±m 2 X (m 2 — 0, 1, 2, . . .) (18.57) 

As a result, the diffraction pattern will have the form of regularly 
arranged spots, with two integral indices m 1 and m 2 corresponding 
to each of them (Fig. 18.39). 

An identical diffraction pattern is obtained if instead of two 
separate gratings we take one transparent plate with two systems 



420 


Optics 


of mutually perpendicular lines applied on it. Such a plate is a 
two-dimensional periodic structure (a conventional grating is a one- 
dimensional structure). Having measured the angles <p x and <p 2 
determining the positions of the maxima and knowing the wavelength 
X, we can use Eqs. (18.56) and (18.57) to find the periods of the struc- 
ture d x and d a . If the directions in which a structure is periodic 
(for example, directions at right angles to the grating lines) make 
the angle a differing from n/2, the diffraction maxima will be at 
the apices of parallelograms instead of at the apices of rectangles 

(as in Fig. 18.39). In this case, 
the diffraction pattern can be used 
to determine not only the periods 
d x and d 2 , but also the angle a. 

Any two-dimensional periodic 
structures such as a system of 
small apertures or one of opaque 
tiny spheres produce a diffraction 
pattern similar to that shown in 
Fig. 18.39. 

For diffraction maxima to ap- 
pear, it is essential that the period 
of the structure d be greater than 
X. Otherwise, conditions (18.56) and (18.57) can be satisfied only at 
values of m x and m 2 equal to zero (the magnitude of sin <p cannot 
exceed unity). 

Diffraction is also observed in three-dimensional structures, i.e. 
spatial formations displaying periodicity along three directions 
not in one plane. All crystalline bodies are such structures. Their 
period (~10 -10 m), however, is too small for the observation of 
diffraction in visible light. The condition d >X is observed for 
crystals only for X-rays. The diffraction of X-rays from crystals 
was first observed in 1913 in an experiment conducted by the German 
physicists Max von Laue (1879-1959), Walter Friedrich (1883-1968), 
and Paul Knipping (1883-1935). (The idea belonged to von Laue, 
while the other two authors ran the experiment.) 

Let us find the conditions for the formation of diffraction maxima 
from a three-dimensional structure. We position the coordinate 
axes x, y, and z in the directions along which the properties of the 
structure display periodicity (Fig. 18.40). The structure can be repre- 
sented as a collection of equally spaced parallel trains of structural 
elements arranged along one of the coordinate axes. We shall con- 
sider the action of an individual linear train parallel, for instance, 
to the x-axis (Fig. 18.41). Assume that a beam of parallel rays mak- 
ing the angle a 0 with the x-axis falls on the train. Every structural 
element is a source of secondary wavelets. An incident wave arrives 
at adjacent sources with a phase difference of 8 0 = 2nA 0 /X, where 




Diffraction of Light 


421 


A 0 = d t cos a 0 (here is the period of the structure along the 
x-axis). Apart from this, the additional path difference A = d^ cos a 
is produced between the secondary wavelets propagating in directions 
that make the angle a with the x-axis (all such directions lie along 
the generatrices of a cone whose axis is the x-axis). The oscillations 
from different structural elements will be mutually amplified for 
the directions for which 

d t (cos a — cos a 0 ) = ± rrijX (m x = 0, 1, 2 f . . .) (18.58) 

There is a separate cone of directions for each value of and 
along these directions we get maxima of the intensity from one 
individually taken train parallel to the x-axis. The axis of this cone 
coincides with the x-axis. 

The condition of the maximum for a train parallel to the y-axis 
has the form 

d 2 (cos p — cos p 0 ) = ±m 2 X (m 2 =0, 1,2, . . .) (18.59) 

where d 2 = period of the structure in the direction of the y-axis 
P 0 = angle between the incident beam and the y-axis 
P = angle between the y-axis and the directions along which 
diffraction maxima are obtained. 

A cone of directions whose axis coincides with the y-axis corre- 
sponds to each value of m 2 . 

In directions satisfying conditions (18.58) and (18.59) simulta- 
neously, mutual amplification of the oscillations from sources in the 
same plane perpendicular to the z-axis occur (these sources form 
a two-dimensional structure). The directions of the intensity maxima 
produced lie along the lines of intersection of the direction cones, 
of which one is determined by condition (18.58), and the second one 
by condition (18.59). 

Finally, for the train parallel to the z-axis, the directions of the 
maxima are determined by the condition 

d s (cos y — cos Yo) = dt m 3 X (m 3 =0, 1, 2, . . .) (18.60) 

where d z = period of the structure in the direction of the z-axis 
y 0 = angle between the incident beam and the z-axis 
y = angle between the z-axis and the directions along which 
diffraction maxima are obtained. 

As in the preceding cases, a cone of directions whose axis coincides 
with the z-axis corresponds to each value of m 3 . 

In the directions satisfying conditions (18.58), (18.59), and (18.60) 
simultaneously, mutual amplification of the oscillations from all 
the elements forming the three-dimensional structure occurs. As 
a result, diffraction maxima are produced by the three-dimensional 
structure. The directions of these maxima are on the lines of inter- 
section of three cones whose axes are parallel to the coordinate axes. 



422 


Optics 


The conditions 

di (cos a — cos a 0 ) = ± 

d 2 (cos P — cos p o ) = ± m 2 X y (m t = 0,1,2, ...) (18.61) 

d 3 (cos y — cos Yo) = ± m 3 X 

which we have found are called Laue’s formulas. Three integral 
numbers m l7 m 2 , and m 3 correspond to each direction (a, P, y) deter- 
mined by these formulas. The greatest value of the magnitude of the 
difference between cosines is two. Hence, conditions (18.61) can be 
obeyed with values of the numbers m other than zero only provided 
that X does not exceed 2d. 

The angles a, p, and y are not independent. For example, when 
a Cartesian system of coordinates is used, they are related by the 
expression 

cos 2 a + cos 2 p + cos 2 y = 1 (18.62) 

Thus, when a 0 , p o , y 0 , and X are given, the angles a, p, y determining 
the directions of the maxima can be found by solving a system of four 
equations. If the number of equations exceeds the number of un- 
knowns, a system of equations can be solved only when definite 
conditions are observed (only when these conditions are satisfied 
can the three cones intersect one another along a single line). 

The system of equations (18.61) and (18.62) can be solved only 
for certain quite definite wavelengths (X can be considered as a fourth 
unknown whose values obtained from the solution of the system of 
equations are exactly the wavelengths for which maxima are ob- 
served). Generally speaking, only one maximum corresponds to each 
such value of X. Several symmetrically arranged maxima may be 
obtained , however. 

If the wavelength is fixed (monochromatic radiation), the system 
of equations can be made simultaneous by varying the values of 
a 0 , Po* and Yo> i- e * by turning the three-dimensional structure relative 
to the direction of the incident beam. 

We have not treated the question of how rays travelling from 
different structural elements are made to converge to one point on 
a screen. A lens does this for visible light. A lens cannot be used 
for X-rays because the refractive index of these rays in all substances 
is virtually equal to unity. For this reason, the interference of the 
secondary wavelets is achieved by using very narrow beams of rays 
producing spots of a very small size on a screen (or a photographic 
plate) even without a lens. 

The Russian scientist Yuri Vulf (1863-1925) and the British phys- 
icists William Henry Bragg (1862-1942) and his son William Law- 
rence Bragg (1890-1971) showed independently of each other that 
the diffraction pattern from a crystal lattice can be calculated in 



Difraction of Light 


423 


the following simple way. Let us draw parallel equispaced planes 
through the points of a crystal lattice (Fig. 18.42). We shall call 
these planes atomic layers. If the wave falling on the crystal is 
plane, the envelope of the secondary waves set up by the atoms in 
such a layer will also be a plane. Thus, the summary action of the 
atoms in one layer cam be represented in the form of a plane wave 
reflected from an atom-covered surface according to the usual law of 
reflection. 

The plane secondary wavelets reflected from different atomic 
layers are coherent and will interfere with one another like the waves 



Fig* 18.42 


Fig. 18.43 


emitted in the given direction by different slits of a diffraction grat- 
ing. As in the case of a grating, the secondary wavelets will virtually 
destroy one another in all directions except those for which the 
path difference between adjacent wavelets is a multiple of X. In- 
spection of Fig. 18.42 shows that the difference between the paths 
of two waves reflected from adjacent atomic layers is 2d sin 0, 
where d is the period of identity of the crystal in a direction at 
right angles to the layers being considered, and 0 is the angle supple- 
menting the angle of incidence and called the glancing angle of the 
incident rays. Consequently, the directions in which diffraction 
maxima are obtained are determined by the condition 

2d sin 0 = ±/7iX (m = 1,2,...) (18.63) 

This expression is known as the Bragg-Vulf formula. 

The atomic layers in a crystal can be drawn in a multitude of ways 
(Fig. 18.43). Each system of layers can produce a diffraction maxi- 
mum if condition (18.63) is observed for it. Only those maxima have 
an appreciable intensity, however, that are obtained as a result of 
reflections from layers sufficiently densely populated by atoms (for 
instance, from layers / and II in Fig. 18.43). 

We must note that calculations by the Bragg-Vulf formula and 
by Laue’s formulas [see Eqs. (18.61)1 lead to coinciding results. 



424 


Optics 


The diffraction of X-rays from crystals has two principal appli- 
cations. It is used to investigate the spectral composition of X-ra- 
diation (X-ray spectroscopy) and to study the structure of crystals 
(X-ray structure analysis). 

By determining the directions of the maxima obtained in the 
diffraction of the X-radiation being studied from crystals with 

a known structure, we can cal- 
culate the wavelengths. Origi- 
nally, crystals of the cubic sys- 
tem were used to determine 
wavelengths, the spacing of the 
planes being determined from 
the density and relative mo- 
lecular mass of the crystal. 

In the method of structural 
analysis proposed by von Laue, 
a beam of X-rays is directed onto 
a stationary monocrystal. The 
radiation contains a wavelength 
at which condition (18.63) is 
satisfied for each system of layers 
sufficiently densely populated by atoms. Consequently, we obtain 
a collection of black spots on a photographic plate placed behind 
the crystal (after development). The mutual arrangement of the 
spots reflects the symmetry of the crystal. The distances between 



Fig. 18.44 



Fig. 18.45 

the spots and their intensities allow us to find the arrangement 
of the atoms in a crystal and their spacing. Figure 18.44 shows 
a Laue diffraction pattern of beryl (a mineral of the silicate group). 

The method of structural analysis developed by the Dutch phys- 
icist Peter Debye and the Swiss physicist Paul Scherrer uses mono- 
chromatic X-radiation and polycrystalline specimens. The substance 
being studied is ground into a powder, and the latter is pressed into 


Diffraction of Light 


425 


a wire-shaped specimen. The specimen is put along the axis of 
a cylindrical chamber on whose side surface a photographic film 
is placed (Fig. 18.45). Among the enormous number of chaotically 
oriented minute crystals, there will always be a multitude of such 
ones for which condition (18.63) will be observed, the diffracted ray 
being in the most diverse planes for different crystals. As a result, 
for each system of atomic layers and each value of m, we get not 



Fig. 18.46 


one direction of a maximum, but a cone of directions whose axis 
coincides with the direction of the incident beam (see Fig. 18.45). 
The pattern obtained on the film (a Debye powder pattern) has the 
form shown in Fig. 18.46. Each pair of symmetrically arranged 
lines corresponds to one of the diffraction maxima satisfying con- 
dition (18.63) at a certain value of m. The structure of the crystal 
can be determined by decoding the X-ray pattern. 


18.8. Resolving Power of an Objective 

Assume that a plane light wave falls on an opaque screen with 
a round aperture of radius b cut out of it. The number of Fresnel 
zones opened by the aperture for point P opposite the centre of the 
aperture at the distance l from it can be found by Eq. (18.13) assum- 
ing that a — oo, r 0 = 6, and b = /. The result is 

(18.64) 

[compare with expression (18.37)1. 

In the same way as for a slit, depending on the value of parameter 
(18.64), we have to do either with the approximation of geometrical 
optics, or Fresnel diffraction, or, finally, Fraunhofer diffraction [see 
expressions (18.36)1. 

We can observe a Fraunhofer diffraction pattern from a round 
aperture on a screen in the focal plane of a lens placed behind the 
aperture by directing a plane light wave onto the aperture. This 
pattern has the form of a central bright spot surrounded by alter- 
nating dark and bright rings (Fig. 18.47). The corresponding cal- 
culations show that the first minimum is at the angular distance 


426 


Optics 


from the centre of the diffraction pattern of 

<Pmrn = arcsin 1.22-^- (18.65) 

where D is the diameter of the aperture (compare with Eq. (18.28)). 
If D X, we may consider that 

<p m ,„=1.22-i (18.66) 

The major part (about 84 per cent) of the light flux passing through 
the aperture gets into the region of the central bright spot. The 

intensity of the first bright ring is only 
1.74 per cent, and of the second, 0.41 per 
cent of the intensity of the central spot. The 
intensity of the other bright rings is still 
smaller. For this reason, in a first approx- 
imation, we may consider that the dif- 
fraction pattern consists of only a single 
bright spot with an angular radius deter- 
mined by Eq. (18.65). This spot is in essence 
the image of an infinitely remote point 
source of light (a plane light wave falls on 
the aperture). 

The diffraction pattern does not depend 
on the distance between the aperture and the 
lens. In particular, it will be the same when 
the edges of the aperture are made to coincide with the edges of 
the lens. It thus follows that even a perfect lens cannot produce an 
ideal optical image. Owing to the wave nature of light, the image 
of a point produced by the lens has the form of a spot that is the 
central maximum of a diffraction pattern. The angular dimension 
oi this spot diminishes with an increasing diameter of the lens 
mount D. 

With a very small angular distance between two points, their 
images obtained with the aid of an optical instrument will be super- 
posed and will produce a single luminous spot. Hence, two very 
close points will not be perceived by the instrument separately or, 
as we say, will not be resolved by the instrument. Consequently, 
no matter how great the image is in size, the corresponding details 
will not be seen on it. 

Let 6^ stand for the smallest angular distance between two points 
at which they can still be resolved by an optical instrument. The 
reciprocal of 6\|? is called the resolving power of the instrument: 




-' 22 $ 0 1 . 22 $ sin ? 


Fig. 18.47 


(18.67) 


Diffraction of Light 


427 


Let us find the resolving power of the objective of a telescope or 
camera when very remote objects are being looked at or photographed. 
In this condition, the rays travelling into the objective from each 
point of the object may be considered parallel, and we can use for- 
mula (18.65). According to the Rayleigh criterion, two close points 
will still be resolved if the middle of the central diffraction maximum 
for one of them coincides with the edge of the central maximum 
(i.e. with the first minimum) for the second one. A glance at Fig. 18.48 



shows that this will occur if the angular distance between the points 
6vJ? will equal the angular radius given by Eq. (18.65). The diameter 
of the objective mount D is much greater than the wavelength X. 
We may therefore consider that 

6* = 1.22 A 


Hence, 




D 

1.22X 


(18.68) 


It can be seen from this formula that the resolving power of an objec- 
tive grows with its diameter. 

The diameter of the pupil of an eye at normal illumination is 
about 2 mm. Using this value in Eq. (18.68) and taking X = 0.5 X 
X 10“ 3 mm, we get 

1.22 x 0,5 =0.305 X 10" 3 rad ^ 1 ' 

Thus, the minimum angular distance between points at which the 
human eye still perceives them separately equals one angular minute. 
It is interesting to note that the distance between adjacent light- 
sensitive elements of the retina corresponds to this angular distance. 


18.9. Holography 

Holography (i.e. ‘‘complete recording”, from the Greek “holos” 
meaning “the whole” and “grapho” — “write”) is a special way of 
recording the structure of the light wave reflected by an object on 



428 


Optics 



a photographic plate. When this plate (a hologram) is illuminated 
with a beam of light, the wave recorded on it is reconstructed in 
practically its original form, so that when the eye perceives the 
reconstructed wave, the visual sensation is virtually the same as 
it would be if the object itself were observed. 

Holography was invented in 1947 by the British physicist Dennis 
Gabor. The complete embodiment of Gabor’s idea became possible, 

however, only after the ap- 
pearance in 1960 of light sources 
having a high degree of coher- 
ence — lasers. Gabor’s initial 
arrangement was improved by 
the American physicists Em- 
met Leith and Juris Upat- 
nieks, who obtained the first 
laser holograms in 1963. The 
Soviet scientist Yuri Denisyuk 
in 1962 proposed an original 
method of recording holograms 
on a thick-layer emulsion. 
This method, unlike holograms 
on a thin-layer emulsion, pro- 
duces a coloured image of the 
object. 

We shall limit ourselves to 
an elementary consideration 
of the method of recording 
holograms on a thin-layer 
emulsion. Figure 18.49a con- 
tains a schematic view of an 
arrangement for recording ho- 
lograms, and Fig. 18.496 a sche- 
matic view of reconstruction 
of the image. The light beam 
emitted by the laser, expanded 
by a system of lenses, is split into two parts. One part is reflected 
by the mirror to the photographic plate forming the so-called refer- 
ence wave 1 . The second part reaches the plate after being reflected 
from the object; it forms object beam 2 . Both beams must be coher- 
ent. This requirement is satisfied because laser radiation has a high 
degree of spatial coherence (the light oscillations are coherent over 
the entire cross section of a laser beam). The reference and object 
beams superpose and form an interference pattern that is recorded 
by the photographic plate. A plate exposed in this way and developed 
is a hologram. Two beams of light participate in forming the holo- 





Diffraction of Light 


429 


gram. In this connection, the arrangement described above is called 
two-beam or split-beam holography. 

To reconstruct the image, the developed photographic plate is 
put in the same place where it was in recording the hologram, and 
is illuminated with the reference beam of light (the part of the laser 
beam that illuminated the object in recording the hologram is now 
stopped). The reference beam diffracts on the hologram, and as a 
result a wave is produced having exactly the same structure as the 
one reflected by the object. This wave produces a virtual image of 
the object that is seen by the observer. In addition to the wave form- 
ing the virtual image, another wave is produced that gives a real 
image of the object. This image is pseu- 
doscopic; this means that it lias a relief 
which is the opposite of the relief of the 
object — the convex spots are replaced by 
concave ones, and vice versa. 

Let us consider the nature of a holo- 
gram and the process of image reconstruc- 
tion. Assume that two coherent parallel 
beams of light rays fall on the photo- 
graphic plate, with the angle between 
the beams (Fig. 18.50). Beam 1 is the 
reference one, and beam 2 , the object 
one (the object in the given case is an infinitely remote point). 
We shall assume for simplicity that beam 1 is normal to the plate. 
All the results obtained below also hold when the reference beam 
falls at an angle, but the formulas will be more cumbersome. 

Owing to the interference of the reference and object beams, a 
system of alternating straight maxima and minima of the intensity 
is formed on the plate. Let points A and B correspond to the middles 
of adjacent interference maxima. Hence, the path difference A' 
equals X. Examination of Fig. 18.50 shows that A' = d sin \|>; hence, 

ds int|) = X (18.69) 

Having recorded the interference pattern on the plate (by exposure 
and developing), we direct reference beam 1 at it. For this beam, 
the plate plays the part of a diffraction grating whose period d 
is determined by Eq. (18.69). A feature of this grating is the cir- 
cumstance that its transmittance changes in a direction perpendic- 
ular to the “lines” according to a cosine law (in the gratings treated 
in Sec. 18.6 it changed in a jump: gap-dark -gap-dark, etc.). The 
result of this feature is that the intensity of all the diffraction maxi- 
ma of orders higher than the first one virtually equals zero. 

When the plate is illuminated with the reference beam (Fig. 18.51), 
a diffraction pattern appears whose maxima form the angles <p 
with a normal to the plate. These angles are determined by the 




430 


Optics 


condition 

d sin (p — mX (m — 0, ±1) (18.70) 

[compare with formula (18.41 )J. The maximum corresponding to 
m = 0 is on the continuation of the reference beam. The maximum 
corresponding to m = +1 has the same direction as object beam 2 
did during the exposure [compare Eqs. (18.69) and (18.70)]. In 
addition, a maximum corresponding to m = —1 appears. 

It can be shown that the result we have obtained also holds when 
object beam 2 consists of diverging rays instead of parallel ones. 
The maximum corresponding tom — +1 has the nature of diverging 
beam of rays 2' (it produces a virtual image of the point from which 
rays 2 emerged during the exposure); the maximum corresponding 
tom = — 1, on the other hand, has the nature of a converging beam 

of rays 2 n (it forms a real image of 
the point which rays 2 emerged from 
during the exposure). 

In recording the hologram, the plate 
is illuminated by reference beam 1 
and numerous diverging beams 2 re- 
flected by different points of the object. 
An intricate interference pattern is 
formed on the plate as a result of 
superposition of the patterns produced 
by each of the beams 2 separately. 
When the hologram is illuminated with 
reference beam 7, all beams 2 are reconstructed, i.e. the complete 
light wave reflected by the object (m = +1 corresponds to it). Two 
other waves appear in addition to it (corresponding to m = 0 and 
m = — 1). But these waves propagate in other directions and do not 
hinder the perception of the wave producing a virtual image of the 
object (see Fig. 18.49). 

The image of an object produced by a hologram is three-dimen- 
sional. It can be viewed from different positions. If in recording 
a hologram close objects concealed more remote ones, then by moving 
to a side we can look behind the closer object (more exactly, behind 
its image) and see the objects that had been concealed previously. 
The explanation is that when moving to a side we see the image 
reconstructed from the peripheral part of the hologram onto which 
the rays reflected from the concealed objects also fell during the 
exposure. When looking at the images of close and far objects, we 
have to accommodate our eyes as when looking at the objects them- 
selves. 

If a hologram is broken into several pieces, then each of them when 
illuminated will produce the same picture as the original hologram. 
But the smaller the part of the hologram used to reconstruct the 




Diffraction of Light 


431 


image, the lower is its sharpness. This is easy to understand by 
taking into account that when the number of lines of a diffraction 
grating is reduced, its resolving power diminishes [see Eq. (18.55)]. 

The possible applications of holography are very diverse. A far 
from complete list of them includes holographic motion pictures 
and television, holographic microscopes, and control of the quality 
of processing articles. The statement can be encountered in publi- 
cations on the subject that holography can be compared as regards 
its consequences with the setting up of radio communication. 


CHAPTER 19 POLARIZATION OF LIGHT 


19.1. Natural and Polarized Light 


We remind our reader that light is called polarized if the direc- 
tions of oscillations of the light vector in it are brought into order 
in some way or other (see Sec. 16.1). In natural light, oscillations 
in various directions rapidly and chaotically replace one another. 

Let us consider two mutually perpendicular electrical oscillations 
occurring along the axes x and y and differing in phase by 6: 

E x = A t cos at, E y = A t cos (©£ -(- 6) (19*1) 


The resultant field strength E is the vector sum of the strengths E* 
and E v (Fig. 19.1). The angle <p between the directions of the vec- 
tors E and E x is determined by the expres- 



sion 


tan (p = -Tr- 


A 2 cos (a)f4-6) 


A 1 cos (ot 


(19.2) 


If the phase difference 6 undergoes random 
chaotic changes, then the angle cp, i.e. the 
direction of the light vector E, will experience 
intermittent disordered changes too. Accord- 
ingly, natural light can be represented as the superposition of 
two incoherent electromagnetic waves polarized in mutually perpen- 
dicular planes and having the same intensity. Such a represen- 
tation greatly simplifies the consideration of the transmission of 
natural light through polarizing devices. 

Assume that the light waves E 
equal to zero or jt. Hence, according to Eq. (19.2), 


x and E v are coherent, with 6 


tan <p= ± -x 5- — c° n st 

A i 

Consequently, the resultant oscillation occurs in a fixed direction— 
the wave is plane-polarized. 

When A x = A t , and 6 = ±n/2, we have 


tan <p = =Ftan c ot 

[cos (( ot ± Jt/2) = =psin c ot]. It thus follows that the plane of 
oscillations rotates about the direction of the ray with an angular 
velocity equal to the frequency of oscillation co. The light in this 
case will be circularly polarized. 



Polarization of Light 


433 


To find the nature of the resultant oscillation with an arbitrary 
constant value of 6, let us take into account that quantities (19.1) 
are the coordinates of the tail of the resultant vector E (Fig. 19.2). 
We know from our treatment of oscillations (see Sec. 7.9 of Vol. I, 
p. 206 et seq.) that two mutually perpendicular harmonic oscilla- 
tions of the same frequency produce motion along an ellipse when 
summated (in particular, motion along a straight line or a circle 
may be obtained). Similarly, a point with the coordinates determined 
by Eqs. (19.1), i.e. the tail of vector E, 
travels along an ellipse. Consequently, 
two coherent plane-polarized light waves 
whose planes of oscillations are mutually 
perpendicular produce an elliptically 
polarized light wave when superposed 
on each others At a phase difference of 
zero or n, the ellipse degenerates into a 
straight line, and plane-polarized light 
is obtained. At 6 = ±ji/2 and equality 
of the amplitude of the waves being 
added, the ellipse transforms into a 
circle — circularly polarized light is Fig- 19-2 

obtained. 

Depending on the direction of rotation of the vector E, right and 
left elliptical and circular polarizations are distinguished. If with 
respect to the direction opposite that of the ray the vector E rotates 
clockwise, the polarisation is called right, and in the opposite case 
it is left. 

The plane in which the light vector oscillates in a plane-polarized 
wave will be called the plane of oscillations. For historical reasons, 
the term plane of polarization was applied not to the plane in which 
the vector E oscillates, but to the plane perpendicular to it. 

Plane-polarized light can be obtained from natural light with 
the aid of devices called polarizers. These devices freely transmit 
oscillations parallel to the plane which we shall call the polarizer 
plane and completely or partly retain the oscillations perpendicular 
to this plane. We shall apply the adjective imperfect to a polarizer 
that only partly retains oscillations perpendicular to its plane. 
We shall apply the term “polarizer” for brevity to a perfect polarizer 
that completely retains the oscillations perpendicular to its plane 
and does not weaken the oscillations parallel, to its plane. 

Light is produced at the outlet from an imperfect polarizer in 
which the oscillations in one direction predominate over the oscil- 
lations in other directions. Such light is called partly polarized. 
It can be considered as a mixture of natural and plane-polarized 
light. Partly polarized light, like natural light, can be represented 
in the form of a superposition of two incoherent plane-polarized 




434 


Opttc» 


waves with mutually perpendicular planes of oscillations. The 
difference is that for natural light the intensity of these waves is 
the same, and for partly polarized light it is different. 

If we pass partly polarized light through a polarizer, then when 
the device rotates about the direction of the ray, the intensity of 
the transmitted light will change within the limits from / max to 
/min* The transition from one of these values to the other one will 
occur upon rotation through an angle of n/2 (during one complete 
revolution both the maximum and the minimum intensity will be 
reached twice). The expression 




fmax — fmln 
fmax + fmln 


(19.3) 


is known as the degree of polarization. For plane-polarized light, 
/ mln = 0, and P = 1. For natural light, / max = / mln , and P = 0. 


i Plane 
r'ofpolariier 



I 

Fig. 19.3 


Plane 



The concept of the degree of polarization cannot be applied to ellip- 
tically polarized light (in such light the oscillations are com- 
pletely ordered, so that the degree of polarization always equals 
unity). 

An oscillation of amplitude A occurring in a plane making the 
angle <p with the polarizer plane can be resolved into two oscillations 
having the amplitudes A\\ = A cos q> and A ± = A sin cp (Fig. 19.3; 
the ray is perpendicular to the plane of the drawing). The first 
oscillation will pass through the device, the second will be retained. 
The intensity of the transmitted wave is proportional to Af\ = 
= A 2 cos 2 <p, i.e. is / cos 2 (p, where / is the intensity of the oscillation 
of amplitude A. Consequently, an oscillation parallel to the plane 
of the polarizer carries along a fraction of the intensity equal to 
cos 2 <p. In natural light, all the values of <p are equally probable. 
Therefore, the fraction of the light transmitted through the polarizer 
will equal the average value of cos 2 qp, i.e. one-half. When the 
polarizer is rotated about the direction of a natural ray, the intensity 
of the transmitted light remains the same. What changes is only 



Polarization of Light 


435 


the orientation of the plane of oscillations of the light leaving the 
device. 

Assume that plane-polarized light of amplitude A 0 and intensity 
7 0 falls on a polarizer (Fig. 19.4). The component of the oscillation 
having the amplitude A = A 0 cos <p, where (p is the angle between 
the plane of oscillations of the incident light and the plane of the 
polarizer, will pass through the device. Hence, 
the intensity of the transmitted light 7 is deter- 
mined by the expression 

I = I 0 cos 2 q> (19.4) 

Relation (lft.4) is known as Malus’s law. It was 
first formulated by the French physicist Etienne 
Malus (1775-1812). 

Let us put two polarizers whose planes make 
the angle <p in the path of a natural ray. 

Plane-polarized light whose intensity 7 0 is 
half that of natural light 7 nat will emerge from 
the first polarizer. According to Malus’s law, 
light having an intensity of 7 0 cos 2 <p will emerge from the second 
polarizer. The intensity of the light transmitted through both 
polarizers is 



/ = y ^nat COS 2 <p (19.5) 

The maximum intensity equal to y 7 nat is obtained at <p = 0 (the 

polarizers are parallel). At <p == ji/ 2, the intensity is zero — crossed 
polarizers transmit no light. 

Assume that elliptically polarized light falls on a polarizer. The 
device transmits the component Ejj of the vector E in the direction 
of the plane of the polarizer (Fig. 19.5). The maximum value of this 
component is reached at points 7 and 2. Hence, the amplitude of 
the plane-polarized light leaving the device equals the length of 01'. 
Rotating the polarizer around the direction of the ray, we shall 
observe changes in the intensity ranging from 7 max (obtained when 
the plane of the polarizer coincides with the semimajor axis of the 
ellipse) to 7 min (obtained when the plane of the polarizer coincides 
with the semiminor axis of the ellipse). The intensity of light for 
partly polarized light will change in the same way upon rotation 
of the polarizer. For circularly polarized light, rotation of the polar- 
izer is not attended (as for natural light) by a change in the intensity 
of the light transmitted through the device. 



436 


Optiet 


19.2. Polarization in Reflection 
and Refraction 

If the angle of incidence of light on the interface between two 
dielectrics (for example, on the surface of a glass plate) differs from 
zero, the reflected and refracted rays will be partly polarized*. 
Oscillations perpendicular to the plane of incidence predominate 
in the reflected ray (in Fig. 19.6 these oscillations are denoted by 

points), and oscillations parallel to the 
plane of incidence predominate in the 
refracted ray (they are depicted in the 
figure by double-headed arrows). The 
degree of polarization depends on the 
angle of incidence. 

Let 0Br stand for the angle satisfying 
the condition 

tan 0Br = 7*12 (19.6) 

(n 12 is the refractive index of the second 
medium relative to the first one). At an 
angle of incidence 0 X equal to 0Bn the 
reflected ray is completely polarized (it 
contains only oscillations perpendicular 
to the plane of incidence). The degree of polarization of the refracted 
ray at an angle of incidence equal to 0B r reaches its maximum value, 
but this ray remains polarized only partly. 

Relation (19.6) is known as Brewster’s law, in honour of its dis- 
coverer, the British physicist David Brewster (1781-1868), and the 
angle 0 Br is called Brewster’s angle. It is easy to see that when light 
falls at Brewster’s angle, the reflected and refracted rays are mutu- 
ally perpend icular . 

The degree of polarization of the reflected and refracted rays for 
different angles of incidence can be obtained with the aid of Fres- 
nel’s formulas. The latter follow from the conditions imposed on 
an electromagnetic field at the interface between two dielectrics**. 
These conditions include the equality of the tangential components 
of the vectors E and H, and also the equality of the normal compo- 
nents of the vectors D and B at both sides of the interface (for one 
side the sum of the relevant vectors for the incident and reflected 



Fig. 19.6 


+ Elliptically polarized light is obtained upon reflection from a conducting 
surface (for example, from the surface of a metal). 

** Fresnel obtained these formulas on the basis of the notions of light as of 
elastic waves propagating in ether. 




Polarization of Light 


437 


waves must be taken, and for the other, the vector for the refracted 
wave). 

Fresnel’s formulas establish the relations between the complex 
amplitudes of the incident, reflected, and refracted waves. We remind 
our reader that by the complex amplitude A is meant the expres- 
sion Ae ia , where A is the conventional amplitude, and a is the 
initial phase of the oscillations. Hence, the equality of two complex 
amplitudes signifies the equality of both the conventional amplitudes 
and the initial phases of the two oscillations: 

Ai = A 2 -+ A s = A 2 ; a* = a 2 (19-7) 

When the complex amplitudes differ in sign, the conventional ones 
are the same, while the initial phases differ by n ( e in = — 1): 

A t = — A 2 ->- = A 2 ; a, = ctj + n (19.8) 


Let us represent the incident wave in the form of a superposition 
of two incoherent waves in one of which the oscillations occur in 

the plane of incidence, and in the other, are perpendicular to this 

plane. Let us denote the complex amplitude of the first wave by ^4|| t 
and of the second by We shall proceed similarly with the 
reflected and refracted waves. We shall use the same symbols for 
the amplitudes of the reflected waves, adding one prime, and the 
same symbols for the amplitudes of the refracted waves, adding 
two primes. Thus, 

j 4 (I and A ± = amplitudes of the incident waves 

A'\\ and A\ = amplitudes of the reflected waves 

A\\ and A\ = amplitudes of the refracted waves 


Fresnel’s formulas have the following form*: 

K- _ A tan (9 X — 9 a ) 

Alt tan(0! + 0 2 ) 




K’ _ _ A sin (0! — 8 t ) 
x 1 sintfii + O,) 


An — A\ 


2 sin 0 a cos 0 t 


N sin (0| + 0 2 ) cos (0j — 0 t ) 

% 2 sin 0 a cos 0 X 

sin(^ + 0 t j 


(19.9) 


(Ox is the angle of incidence, and 0 2 is the angle of refraction of the 
light wave). We must underline the fact that formulas (19.9) estab- 
lish the relations between the complex amplitudes at the interface 


• Fresnel’s formulas are customarily written without “caps” over the ampli- 
tudes. To underline the fact that we are dealing with complex amplitudes, 
however, we found it helpful to write the amplitudes with the u caps . 



438 


Optics 


between dielectrics, i.e. at the point of incidence of a ray on this 
interface. 

It can be seen from the last two of formulas (19.9) that the signs 
of the complex amplitudes of the incident and refracted waves at any 
values of the angles 0 X and 0 a are the same (the sum of 0 X and 0 t 
cannot exceed ji). This signifies that when penetrating into the 
second medium, the phase of the wave does not undergo a jump. 

In dealing with the phase relations between an incident and a 
reflected wave, we must take into account that for a wave polarized 




Fig. 19.7 

perpendicularly to the plane of incidence, the coincidence of the 
signs of A x and A x corresponds to the absence of a jump in the 
phase in reflection (Fig. 19.7a). For a wave that is polarized in the 
plane of incidence, on the other hand, a jump in the phase is absent 
when the signs of A\\ and A\\ are opposite (Fig. 19.76). 

The phase relations between the reflected and incident waves 
depend on the relation between the refractive indices n x and rt t 
of the first and second media, and also on the relation between the 
angle of incidence Q x and Brewster’s angle 0Br (we remind our reader 
that when Q x == 0 Br , the sum of the angles 0 X and 0 a is n/2). Table 19.1 
gives the results following from the first two of formulas (19.9) in 
four possible cases. It follows from the table that for incidence at an 
angle less than Brewster’s angle, reflection from an optically denser 
medium is attended by a jump in phase of n; reflection from an opti- 
cally less dense medium occurs without a change in phase. This 
result for 0 X = 0 was obtained in Sec. 16.3. When 0 X 2>0 Br » the 
phase relations for both wave components are different. 

We obtain from the first of formulas (19.9) that when Q x + 0* = 
== ji/ 2, i.e. at 0 X = 0 Br * the amplitude A j| vanishes. Consequently, 
only oscillations perpendicular to the plane of incidence are present 
in the reflected wave — the latter is completely polarized. Thus, 
Brewster’s law directly follows from Fresnel’s formulas. 



Polarization of Light 


439 


Table 19.1 



01 < ®Br 
(01+9* <«/2) 

®l > ®Br 
(01 + 0* >"/2) 

n 2 > n x 

e 1 >0. 

The signs of A ' 8 and Am are 
the same (a phase jump by n) 

The sign of A' x is opposite to 
that of A' x (a phase jump 
by n) 

The sign of A [. is opposite to 
that of A || (no phase jump) 
The sign of A' ± is opposite to 
that of A ± (a phase jump 
by n) 

n t < 

01 <0* 

The sign of A 'y is opposite to 
that of A || (no phase jump) 
The signs of A' ± and A ± are 
the same (no phase jump) 

The signs of A ^ and A* are 
the same (a phase jump oy n) 

The signs of A' ± and A ± are 
the same (no phase jump) 


At small angles of incidence, the sines and tangents in formulas 
(19.9) may be replaced by the angles themselves, and the cosines 
may be assumed equal to unity. In addition, in this case we may 
consider that = n 12 9 2 (this follows from the law of refraction 
after the sines are replaced with the relevant angles). As a result, 
Fresnel’s formulas for small angles of incidence acquire the form 



A, 

e,-0, _ 
^1 + ^2 


n lt — l 
n„+ 1 

^4 

II 

T ©X — 0* 

Al 0i+0. 

• = 

A ”i»— 1 

"1S+ 1 

A: — 

An 


An- 

2 


H n 

^1 + 

^11 

«i*+l 

A" __ 

A 

20, 

A , 

2 



01 + 0. 


*12+1 


Squaring Eqs. (19.10) and multiplying the expressions obtained 
by the refractive index of the relevant medium, we get relations 
between the intensities of the incident, reflected, and refracted rays 
for small angles of incidence [see expression (16.9)]. Here, for exam 
pie, the intensity of the reflected light /' can be calculated as the sum 
of the intensities of both components I\\ and I' x because these com- 
ponents are not coherent in natural light [the intensities instead of 
the amplitudes are summated for incoherent waves, see Eq. (17.1)1. 
As a result, we get 


/' 



* 12—1 \ 2 
* 12+1 ) ’ 


(^tt) 


From these formulas, we get Eqs. (16.33) and (16.34) for p and t. 











440 


Optics 


19.3, Polarization in Double Refraction 

When light passes through all transparent crystals except for 
those belonging to the cubic system, a phenomenon is observed 
called double refraction*. It consists in that a ray falling on a crystal 
is split inside the latter into two rays propagating, generally speak- 
ing, with different velocities and in different directions. 

Doubly refracting (or birefringent) crystals are divided into 
uniaxial and biaxial ones. In uniaxial crystals, one of the refracted 
rays obeys the conventional law of refraction, in particular it is 
in the same plane as the incident ray and a normal to the refracting 

surface. This ray is called an ordinary ray and 
is designated by the symbol o. For the other 
^tt *■ e ray, called an extraordinary ray (designated 
j by e) y the ratio of the sines of the angle of 
incidence and the angle of refraction does not 
remain constant when the angle of incidence 
varies. Even upon normal incidence of light 
Fig. 19.8 on a crystal, an extraordinary ray, generally 

speaking, deviates from a normal (Fig. 19.8). 
In addition, an extraordinary ray does not lie, as a rule, in the same 
plane as the incident ray and a normal to the refracting surface. 
Examples of uniaxial crystals are Iceland spar, quartz, and tour- 
maline. In biaxial crystals (mica, gypsum), both rays are extra- 
ordinary — the refractive indices for them depend on the direction 
in the crystal. In the following, we shall be concerned only with 
uniaxial crystals. 

Uniaxial crystals have a direction along which ordinary and 
extraordinary rays propagate without separation and with the same 
velocity**. This direction is known as the optical axis of the crystal. 
It must be borne in mind that an optical axis is not a straight line 
passing through a point of a crystal, but a definite direction in 
the crystal. Any straight line parallel to the given direction is an 
optical axis of the crystal. 

A plane passing through an optical axis is called a principal section 
or a principal plane of the crystal. Customarily, the principal section 
passing through the light ray is used. 

Investigation of the ordinary and extraordinary rays shows that 
they are both completely polarized in mutually perpendicular direc- 
tions (see Fig. 19.8). The plane of oscillations of the ordinary ray is 
perpendicular to a principal section of the crystal. In the extra- 

* Double refraction was first observed in 1669 by the Danish scientist Erasm 
Bartholin (1625-1698) for Iceland spar (a variety of calcium carbonate CaCO s — 
crystals of the hexagonal system). 

+* Biaxial crystals have two such directions. 





Polarization of Light 


441 


ordinary ray, the oscillations of the light vector occur in a plane 
coinciding with a principal section. When they emerge from the 
crystal, the two rays differ from each other only in the direction of 
polarization so that the terms “ordinary” and “extraordinary” have 
a meaning only inside the crystal. 

In some crystals, one of the rays is absorbed to a greater extent 
than the other. This phenomenon is called dichroism. A crystal of 
tourmaline (a mineral of a complex composition) displays very 
great dichroism in visible rays. An ordi- 
nary ray is virtually completely absorbed 
in it over a distance of 1 mm. In crystals 
of iodoquinine sulphate, one of the rays is 
absorbed over a path of about 0.1 mm. 

This circumstance has been taken advan- 
tage of for manufacturing a polarizing 
device called a polaroid. It is a celluloid 
film into which a great number of identi- 
cally oriented minute crystals of iodoqui- 
nine sulphate have been introduced. 

Double refraction is explained by the ani- 
sotropy of crystals. In crystals of the non- 
cubic system, the permittivity e depends on 
the direction. In uniaxial crystals, e in 
the direction of an optical axis and in 
directions perpendicular to it has different values e\\ and e ± . 
In other directions, e has intermediate values. According to Eq. (16.3) 
n = Y~e. It thus follows from the anisotropy of e that different 
values of the refractive index n correspond to electromagnetic waves 
with different directions of the oscillations of the vector E. Therefore, 
the velocity of the light waves depends on the direction of oscilla- 
tions of the light vector E. 

In an ordinary ray, the oscillations of the light vector occur in 
a direction perpendicular to a principal section of the crystal (in 
Fig. 19.9 these oscillations are depicted by dots on the relevant ray). 
Therefore, with any direction of an ordinary ray (three directions i, 
2, and 3 are shown in the figure), the vector E makes a right angle 
with an optical axis of the crystal, and the velocity of the light wave 
will be the same, equal to v Q = c/|/ e ± . Depicting the velocity of an 
ordinary ray in the form of lengths laid off in different directions, 
we shall get a spherical surface. Figure 19.9 shows the intersection 
of this surface with the plane of the drawing. A picture such as that 
in Fig. 19.9 is observed in any principal section, i.e. in any plane 
passing through an optical axis. Let us imagine that a point source 
of light is placed at point O inside a crystal. Hence, the sphere 
which we have constructed will be the wave surface of ordinary 
rays. 


Optical axis 
Y of crystal 
7 



442 


Optics 


The oscillations in an extraordinary ray take place in a principal 
section. Therefore, for different rays, the directions of oscillations 
of the vector E (in Fig. 19.9 these directions are depicted by double- 
headed arrows) make different angles a with an optical axis. For 
ray 7, the angle a is ji/2, owing to which the velocity is u Q = c/|/ e x , 
for ray 2, the angle a = 0, and the velocity is v e = c/Y e n- For 
ray 2, the velocity has an intermediate value. We can show that the 
wave surface of extraordinary rays is an ellipsoid of revolution. At 
places of intersection with an optical axis of the crystal, this ellip- 
soid and the sphere constructed for the ordinary rays come into con- 
tact. 

Uniaxial crystals are characterized by a refractive index of an 
ordinary ray equal to n 0 = c/v 01 and a refractive index of an extra- 




ordinary ray perpendicular to an optical axis equal to n e — c/y e . 
The latter quantity is called simply the refractive index of an extra- 
ordinary ray. 

Depending on which of the velocities, v Q or v ei is greater, positive 
and negative uniaxial crystals are distinguished (Fig. 19.10). For 
positive crystals, v e < v 0 (this means that n e >n Q ). For negative 
crystals, v e >u 0 (n e < n 0 ). It is simple to remember what crystals 
are called positive and what negative. For positive crystals, the 
ellipsoid of velocities is extended along an optical axis reminding 
one of the vertical line in the sign for negative crystals, the 

ellipsoid of velocities is extended in a direction perpendicular to an 
optical axis, reminding one of the sign “ — ”. 

The path of an ordinary and an extraordinary ray in a crystal 
can be determined with the aid of the Huygens principle. Figure 19.11 
depicts wave surfaces of an ordinary and extraordinary rays with 
their centre at point 2 on the surface of the crystal. The construction 
is for the moment of time when the wavefront of the incident wave 
reaches point 1 . The envelopes of all the secondary wavelets (the 



Polarization of Light 


443 


waves whose centres are in the interval between points 1 and 2 
are not shown in the figure) for the ordinary and extraordinary rays 
are evidently planes. The refracted ray o or e emerging from point 2 
passes through the point of con- 
tact of the envelope with the 
relevant wave surface. 

We remind our reader that 
rays are defined as lines along 
which the energy of a light wave 
propagates (see Sec. 16.1). A 
glance at Fig. 19.11 shows that the 
ordinary ray o coincides with 
a normal to the relevant wave 
surface. The extraordinary ray 
e, on the other hand, appreciably 
deviates from a normal to the 
wave surface. 

Figure 19.12 shows three cases 
of the normal incidence of light 
on the surface of a crystal differ- 
ing in the direction of the opti- 
cal axis. In case a, the rays o and 
e propagate along an optical axis 
and therefore travel without sepa- 
rating. Inspection of Fig. 19.12b 
shows that even upon normal 
incidence of light on a refract- 
ing surface, an extraordinary ray may deviate from a normal to 
this surface (compare with Fig. 19.8). In Fig. 19.12c, the optical 
axis of the crystal is parallel to the refracting surface. In this case 
with normal incidence of the light, the ordinary and extraordinary 
rays travel in the same direction, but propagate with different 
velocities. The result is a constantly growing phase difference 
between them. The nature of polarization of the ordinary and 
extraordinary rays in Fig. 19.12 is not indicated. It is the same as 
for the rays depicted in Fig. 19.11. 

19.4. Interference of Polarized Rays 

When two coherent rays polarized in mutually perpendicular 
directions are superposed, no interference pattern with the charac- 
teristic alternation of maxima and minima of the intensity can be 
obtained. Interference occurs only when the oscillations in the inter- 
acting rays occur along the same direction. The oscillations in two 
rays initially polarized in mutually perpendicular directions can be 






)J vv 

JJ 

i o\ 

Axis 

e o 

(a) 

l 1 

e 

L 

mm 

m mm 

El 



*6 / l 

\ 71 

7 (b) e 1 

r 

7 


sj 


xa 

Axis 

X- 1 XJ 



(o) 

Fig. 19.12 


444 


Optics 


brought into one plane by passing these rays through a polarizer 
installed so that its plane does not coincide with the plane of oscil- 
lations of any of the rays. 

Let us see what happens when an ordinary and an extraordinary 
ray emerging from a crystal plate are superposed. Assume that the 
plate has been cut out parallel to an optical axis (Fig. 19.13). With 
normal incidence of the light on the plate, the ordinary and extra- 
ordinary rays will propagate without separating, but with different 


Axis 



(a) W 

Fig. 19.13 


velocities (see Fig. 19.12c). The following path difference appears 
between the rays while they pass through the plate: 

A = (n 0 — n t ) d (19.11) 

or the following phase difference: 

6 = 2 n (19.12) 

(i d is the plate thickness, and k 0 the wavelength in a vacuum). 

Thus, if we pass natural light through a crystal plate cut out 
parallel to the optical axis (Fig. 19.13a), two rays 1 and 2 that are 
polarized in mutually perpendicular planes will emerge from the 
plate*, and between them there will be a phase difference determined 
by Eq. (19.12). Let us place a polarizer in the path of these rays. 
Both rays after passing through the polarizer will oscillate in one 
plane. Their amplitudes will equal the components of the amplitudes 
of rays 1 and 2 in the direction of the plane of the polarizer 
(Fig. 19.13&). 

The rays emerging from the polarizer are produced as a result 
of division of the light obtained from a single source. Therefore, 
they ought to interfere. If rays 1 and 2 are produced as a result 


• In the crystal, ray 1 was extraordinary and could be designated by the sym- 
bol e , and ray 2 was ordinary (o). Upon emerging from the crystal, these rays lost 
their right to be called ordinary and extraordinary. 



Polarization of Light 


445 


of natural light passing through the plate, however, they do not 
interfere. The explanation is very simple. Although the ordinary 
and extraordinary rays are produced by the same light source, they 
contain mainly oscillations belonging to different wave trains emit- 
ted by individual atoms. The oscillations in the ordinary ray are 
predominantly due to the trains whose oscillation planes are close 
to one direction in space, whereas those in the extraordinary ray are 
due to trains whose oscillation planes are close to another direction 
perpendicular to the first one. Since the individual trains are inco- 
herent, the ordinary and extraordinary rays produced from natural 
light, and, consequently, rays 1 and 2 too, are also incoherent. 

Matters are different if plane-polarized light falls on a crystal 
plate. In this case, the oscillations of each train are divided between 
the ordinary and extraordinary rays in the same proportion (depend- 
ing on the orientation of an optical axis of the plate relative to the 
plane of oscillations in the incident ray). Consequently, rays o 
and e, and therefore rays 1 and 2 too, will be coherent and will 
interfere. 


19.5. Passing of Plane-Polarized 

Light Through a Crystal Plate 

Let us consider a crystal plate cut out parallel to an optical axis. 
We saw in the preceding section that when plane-polarized light 
falls on such a plate, the ordinary and extraordinary rays are coher- 
ent. At the entrance to the plate, the phase difference 6 of these 
rays is zero, and at the exit from the plate 

6 = -^2n= (n °~” e)d 2n (19.13) 

[see Eqs. (19.11) and (19.12); we assume that the light falls on the 
plate normally]. 

A plate cut out parallel to an optical axis for which 
(n 0 — n e ) d = mko + ^ 

{m is any integer or zero) is called a quarter-wave plate. An ordinary 
and an extraordinary rays passing through such a plate acquire 
a phase difference equal to n/2 (we remind our reader that the phase 
difference is determined with an accuracy to 2nm). A plate for which 

(n 0 —n e )d = mk 0 + -y- 


is called a half-wave plate, etc. 



446 


Opttes 


Let us see how plane-polarized light passes through a half-wave 
dlate. The oscillation of E in the incident ray occurring in plane P 
produces the oscillation of E 0 of the ordinary ray and the oscillation 
of E e of the extraordinary ray when entering the crystal (Fig. 19.14). 
During the time spent in passing through the plate, the phase differ- 
ence between the oscillations of E 0 and E e changes by jt. Therefore, 
at the exit from the plate, the phase relation between the ordinary 
and extraordinary rays will correspond to the mutual arrangement 
of the vectors E* and Eq (at the entrance to the plate it corresponded 



O 



Fig. 19.15 


to the mutual arrangement of the vectors E e and E 0 ). Consequently, 
the light emerging from the plate will be polarized in plane P\ 
Planes P and P 9 are symmetrical relative to optical axis O of the 
plate. Thus, a half-wave plate turns the plane of oscillations of the 
light passing through it through the angle 2<p (q> is the angle between 
the plane of oscillations in the incident ray and the axis of the plate). 

Now let us pass plane-polarized light through a quarter-wave plate 
(Fig. 19.15). If we arrange the plate so that the angle <p between 
plane of oscillations P in the incident ray and plate axis O is 45 de- 
grees, the amplitudes of both rays emerging from the plate will be 
the same (dichroism is assumed to be absent). The phase shift be- 
tween the oscillations in these rays will be nl 2. Hence, the light emerg- 
ing from the plate will be circularly polarized. At a different value 
of the angle (p, the amplitudes of the rays emerging from the plate 
will be different. Consequently, these rays when superposed form 
elliptically polarized light; one of the axes of the ellipse coincides 
with plate axis O . 

When plane-polarized light is passed through a plate with a frac- 
tional number of waves not coinciding with m + ^ or m + , 

two coherent light waves polarized in mutually perpendicular planes 


Polarization of Light 


447 


will emerge from the plate. Their phase difference is other than 
n/2 and other than ji. Hence, with any relation between the ampli- 
tudes of these waves depending on the angle <p (see Fig. 19.15), ellip- 
tically polarized light will be produced at the exit from the plate, 
and none of the axes of the ellipse will coincide with plate axis 0. 
The orientation of the ellipse axes relative to axis O is determined 
by the phase difference 6, and also by the ratio of the amplitudes, 
i.e. by the angle <p between the plane of oscillations in the incident 
wave and plate axis 0. 

We must note that regardless of the plate thickness, when <p is 
zero or ji/ 2, only one ray will propagate in the plate (in the first case 
an extraordinary ray, in the second case an ordinary one) so that at 
the plate exit the light remains plane-polarized with its plane of 
oscillations coinciding with P . 

If we place a quarter-wave plate in the path of elliptically 
polarized light and arrange its optical axis along one of the 
ellipse axes, then the plate will introduce an additional phase 
difference equal to jt/2. As a result, the phase difference between 
two plane-polarized waves whose sum is an elliptically po- 
larized wave becomes equal to zero or ji, so that the superposition of 
these waves produces a plane-polarized wave. Hence, a properly turned 
quarter-wave plate transforms elliptically polarized light into 
plane-polarized light. This underlies a method by means of which we 
can distinguish elliptically polarized light from partly polarized 
light, or circularly polarized light from natural light. The light being 
studied is passed through a quarter-wave plate and a polarizer placed 
after it. If the ray being studied is elliptically polarized (or circular- 
ly polarized), then by rotating the plate and the polarizer around the 
direction of the ray, we can achieve complete darkening of the field 
of. vision. If the light is partly polarized (or natural), it is impossible 
to achieve extinction of the ray being studied with any position of 
the plate and polarizer. 


19.6. A Crystal Plate 

Between Two Polarizers 

Let us place a plate made from a uniaxial crystal cut out parallel 
to optical axis O between polarizers* P and P* (Fig. 19.16). Plane- 
polarized light of intensity I will emerge from polarizer P. In pas- 
sing through the plate, the light in the general case will become el- 
liptically polarized. When it emerges from polarizer P\ the light 
will again be plane-polarized. Its intensity /' depends on the mutual 


• The second polarizer P r in the direction of ray propagation is also called 
an analyzer. 


448 


Optics 


orientation of the planes of polarizers P and P' and an optical axis 
of the plate, and also on the phase difference 6 acquired by the ordin- 
ary and extraordinary rays when they pass through the plate. 
Assume that the angle cp between the plane of polarizer P and plate 
axis O is n/4. Let us consider two particular cases: the polarizers are 

O 



■ I 1 

v\\\ 

n 

7 ' 


/ 

u 

I 1 

r ' 


o 

Fig. 19.16 


parallel (Fig. 19.17a), and they are crossed (Fig. 19.176). The light 
oscillation leaving polarizer P will be depicted by the vector E in 
plane P. At the entrance to the plate, the oscillation of E will pro- 
duce two oscillations — the oscillation of E 0 (ordinary ray) perpendi- 
cular to the optical axis, and the oscillation of E e (extraordinary ray) 




Fig. 19.17 

parallel to the axis. These oscillations will be coherent; in passing 
through the plate, they acquire the phase difference 6 that is deter- 
mined by the plate thickness and the difference between the refrac- 
tive indices of the ordinary and extraordinary rays. The amplitudes 
of these oscillations are the same and equal 

E 0 = E e -Ecos^ = JL=. (19.14) 

where E is the amplitude of the wave emerging from the first po- 
larizer. 



Polarization .of Light 


449 


The components of the oscillations of E 0 and E e will pass through 
the second polarizer in the direction of plane P '. The amplitudes of 
these components in both cases equal those given by Eq. (19.14) 
multiplied by cos (ji/ 4), i.e. 

E' 0 = E' e =-J (19.15) 

For parallel polarizers (Fig. 19.17a), the phase difference of the 
waves emerging from polarizer P' is 6, i.e. the phase difference ac- 
quired when passing through the plate. For crossed polarizers (Fig. 
19.176), the projections of the vectors E 0 and E e onto the direction of 
P' have different signs. This signifies that an additional phase 
difference equal to n appears apart from the phase difference 6. 

The waves leaving the second polarizer will interfere. The ampli- 
tude E\\ of the resultant wave for parallel polarizers is determined by 
the relation 

£f, = E' 0 2 + E? + 2 E'oE'e cos 6 
and for crossed polarizers by the relation 

E\ — E[ ? + E' e * + 2 E'oE* cos (6 + n) 

Taking Eq. (19.15) into consideration, we can write that 

E\ \ = ± E* + ±Ez + 2x±-E*cos6= ^ ^ (1 +cos 6) = £* cos*-|. 

E\ = ^ E*+ iE 2 + 2x-^£ 2 cos (6 + n) = 

= y & (1 — cos 6) = E 2 sin 2 

The intensity is proportional to the square of the amplitude. 
Hence, 

/,', = / cos* /i = /sin 2 y (19.16) 

Here /(| = intensity of the light emerging from the second polarizer 
when the polarizers are parallel 
I' ± = the same intensity when the polarizers are crossed 
I = intensity of the light that has passed through the first 
polarizer. 

It follows from formulas (19.16) that the intensities /|| and I' x are 
“complementary” — their sum gives the intensity /. In particular, 
when 

8 = 2mn (m =1,2, ...) (19.17) 

the intensity I\\ will equal /, while the intensity Jj. will vanish. 
At values of 


8 = (2m. + 1) J* (m = 0, 1, 2, ...) 


(19.18) 



450 


Optics 


on the other hand, the intensity I[\ will vanish, while the intensity 
I' ± reaches the value I . 

The difference between the refractive indices n Q — n e depends on 
the wavelength of the light X 0 . In addition, X 0 directly enters expres- 
sion (19.13) for 6. Assume that the light falling on polarizer P con- 
sists of radiation of two wavelengths X 1 and X 2 such that 6 for Xj 
satisfies condition (19.17), and for X 2 condition (19.18). In this case 
with parallel polarizers, light of wavelength X x will pass without 
hindrance through the system depicted in Fig. 19.16, whereas light 



0 

(a) (*>) 

Fig. 9.18 


of wavelength X 2 will be made completely extinct. With crossed 
polarizers, light of wavelength X 2 will pass without hindrance, and 
light of wavelength X* will be made completely extinct. Consequent- 
ly, with one arrangement of the polarizers, the colour of the light 
transmitted through the system will correspond to the wavelength 
Xj, and w r ith the other arrangement, to the wavelength X 2 . Such two 
colours are called complementary. When one of the polarizers is 
rotated, the colour continuously changes, varying during each quar- 
ter of a revolution from one complementary colour to the other. 
A change in colour is also observed at q> differing from ji/ 4 (but not 
equal to zero or ji/2), the colours being less saturated, however. 

The phase difference 6 depends on the plate thickness. Hence, if 
a doubly refracting transparent plate placed between polarizers has 
a different thickness at different places, the latter when observed 
from the side of polarizer P ' will seem to be coloured differently. 
When polarizer P' is rotated, these colours change, each of them trans- 
forming into its complementary colour. Let us explain this by the 
following example. Figure 19.18a shows a plate placed between po- 
larizers. The bottom half of the plate is thicker than the top one. 
Assume that the light passing through the plate contains radiation 
of only two wavelengths X 2 and X 2 . Figure 19.186 gives a “view” from 
the side of polarizer P\ At the exit from the crystal plate, each of 



Polarization of Light 


451 


the light components will, generally speaking, be elliptically polar- 
ized. The orientation and the eccentricity of the ellipses for the wave- 
lengths \ and X 2 , and also for different halves of the plate, will 
be different. When the plane of polarizer P' is placed in position 
/>', in the light transmitted through P 9 the wavelength Xj will pre- 
dominate in the top half of the plate and the wavelength X 2 in the 
bottom half. Therefore, the two halves will be coloured differently. 
When polarizer P 9 is placed in position />', the colour of the top 
half will be determined by the light of wavelength X 2 , and of the 
bottom half by the light of wavelength X 1 . Thus, when polarizer 
P 9 is turned through 90 degrees, the two halves of the plate exchange 
colours, as it were. It is quite natural that this will occur only at 
a definite ratio of the thicknesses of the two halves of the plate. 


19.7. Artificial Double Refraction 


External action may cause double refraction to appear in trans- 
parent amorphous bodies, and also in crystals of the cubic system. 
This occurs, in particular, upon the mechanical deformations of 



Fig. 19.19 


Fig. 19.20 


bodies. The difference between the refractive indices of an ordinary 
and an extraordinary ray is a measure of the appearance of optical 
anisotropy. Experiments show that this difference is proportional to 
the stress a at a given point of a body (i.e. to the force per unit area; 
see Sec. 2.9 of Vol. I, p. 64 et seq.): 

n Q — n e = ka (19.19) 

(k is a proportionality constant depending on the properties of the 
substance). 

Let us place glass plate Q between crossed polarizers P and P* 
(Fig. 19.19). As long as the glass is not deformed, such a system trans- 
mits no light. If the plate is subjected to compression, light begins 
to pass through the system, the pattern observed in the transmitted 
rays being speckled with coloured fringes. Each fringe corresponds 




452 


Optics 


to identically deformed spots on the plate. Consequently, the dis- 
tribution of the fringes makes it possible to assess the distribution 
of the stresses inside the plate. This underlies the optical method 
of studying stresses. A model of a component or structural member 
made from a transparent isotropic material (for example, from Plexi- 
glas) is placed between crossed polarizers. The model is subjected 
to the action of loads similar to those which the article itself will 
experience. The pattern observed in transmitted white light makes 
it possible to determine the distribution of the stresses and also to 
estimate their magnitude. 

The appearance of double refraction in liquids and amorphous sol- 
ids under the action of an electric field was discovered by the Scotch 
physicist John Kerr (1824-1907) in 1875. This effect was named the 
Kerr effect after its discoverer. In 1930, it was also observed in ga- 
ses. 

An arrangement for studying the Kerr effect in liquids is shown 
schematically in Fig. 19.20. It consists of a Kerr cell placed be- 
tween crossed polarizers P and P'. A Kerr cell is a sealed vessel con- 
taining a liquid into which capacitor plates have been introduced. 
When a voltage is applied across the plates, a virtually homogeneous 
electric field is set up between them. Under its action, the liquid ac- 
quires the properties of a uniaxial crystal with an optical axis orien- 
ted along the field. 

The resulting difference between the refractive indices n 0 and n c is 
proportional to the square of the field strength E: 

n Q — n e = kE z (19.20) 

The path difference 

A = (n Q — n e ) l = klE 2 

appears between the ordinary and extraordinary rays along the path L 
The corresponding phase difference is 

6 = 2n = 2n ~ IE 1 

The latter expression is conventionally written in the form 

8 =2 nBlE* (19.21) 

where B is a quantity characteristic of a given substance and known 
as the Kerr constant. 

The Kerr constant depends on the temperature of a substance and 
on the wavelength of the light. Among known liquids, nitrobenzene 
(C 6 H 5 N0 2 ) has the highest Kerr constant. 

The Kerr effect is explained by the different polarization of mole- 
cules in various directions. In the absence of a field, the molecules 
are oriented chaotically, therefore a liquid as a whole displays no 



Polarization of Light 


453 


anisotropy. Under the action of a field, the molecules turn so that 
either their electric dipole moments (in polar molecules) or their di- 
rections of maximum polarization (innon-polar molecules) are orient- 
ed in the direction of the field. As a result, the liquid becomes opti- 
cally anisotropic. The thermal motion of the molecules counteracts 
the orienting action of the field. This explains the reduction in the 
Kerr constant with elevation of the temperature. 

The time during which the prevailing orientation of the molecules 
sets in (when the field is switched on) or vanishes (when the field is 
switched off) is about 10' 10 s. Therefore, a Kerr cell placed between 
crossed polarizers can be used as a virtually inertialess light shutter. 
In the absence of a voltage across the capacitor plates, the shutter 
will be closed. When the voltage is switched on, the shutter trans- 
mits a considerable part of the light falling on the first polarizer. 


19.8. Rotation of Polarization Plane 

Natural Rotation. Some substances known as optically active 
ones have the ability of causing rotation of the plane of polarization 
of plane-polarized light passing through them. Such substances in- 
clude crystalline bodies (for example, quartz, cinnabar), pure liquids 
(turpentine, nicotine), and solutions of optically active substances 
in inactive solvents (aqueous solutions of sugar, tartaric acid, etc.). 

Crystalline substances rotate the plane of polarization to the great- 
est extent when the light propagates along the optical axis of the 
crystal. The angle of rotation <p is proportional to the path l tra- 
velled by a ray in the crystal: 

<p = al (19.22) 

The coefficient a is called the rotational constant. It depends on 
the wavelength (dispersion of the ability to rotate). 

In solutions, the angle of rotation of the plane of polarization is 
proportional to the path of the light in the solution and to the con- 
centration of the active substance c : 

cp = [a] cl (19.23) 

Here la] is a quantity called the specific rotational constant. 

Depending on the direction of rotation of the polarization plane, 
optically active substances are divided into right-hand and left- 
hand ones. The direction of rotation (relative to a ray) does not 
depend on the direction of the ray. Consequently, if a ray that has 
passed through an optically active crystal along its optical axis is 
reflected by a mirror and made to pass through the crystal again in 
the opposite direction, then the initial position of the polarization 
nlane is restored. 



454 


Optics 


All optically active substances exist in two varieties — right- 
hand and left-hand. There exist right-hand and left-hand quartz, 
right-hand and left-hand sugar, etc. The molecules or crystals of 
one variety are a mirror image of the molecules or crystals of the other 
one (Fig. 19.21). The symbols C , X, Y> Z, and T stand for atoms or 
groups of atoms (radicals) differing from one another. Molecule b 
is a mirror image of molecule a. If we look at the tetrahedron depicted 
in Fig. 19.21 along the direction CX, then in clockwise circumven- 
tion we shall encounter the sequence ZYTZ for molecule a and ZTYZ 
for molecule b . The same is observed for any of the directions CY , 

CZ, and CT. The alternation of the radicals X, 
y\ Z, T in molecule b is the opposite of their 
alternation in molecule a . Consequently, if, 
for example, a substance formed of molecules a 
is right-hand, then one formed of molecules 
b is left-hand. 

If we place an optically active substance 
(a crystal of quartz, a transparent tray with 
a sugar solution, etc.) between two crossed 
polarizers, then the field of vision becomes bright. To get darkness 
again, one of the polarizers has to be rotated through the angle 
<p determined by expression (19.22) or (19.23). When a solution is 
used, we can determine its concentration c by Eq. (19.23) if we know 
the specific rotational constant [ a 1 of the given substance and the 
length l and have measured the angle of rotation cp. This way of de- 
termining the concentration is used in the production of various sub- 
stances, in particular in the sugar industry (the corresponding in- 
strument is called a saccharimeter). 

Magnetic Rotation of the Polarization Plane. Optically inactive 
substances acquire the ability of rotating the plane of polarization 
under the action of a magnetic field. This phenomenon was discov- 
ered by Michael Faraday and is therefore sometimes called the Fa- 
raday effect. It is observed only when light propagates along the 
direction of magnetization. Therefore, to observe the Faraday effect, 
holes are drilled in the pole shoes of an electromagnet, and a light 
ray is passed through them. The substance being studied is placed 
between the poles of the electromagnet. 

The angle of rotation of the polarization plane <p is proportional 
to the distance l travelled by the light in the substance and to the 
magnetization of the latter. The magnetization, in turn, is propor- 
tional to the magnetic field strength H [see Eq. (7.14)]. We can 
therefore write that 

q> = VIH (19.24) 

The coefficient V is known as the Verdet constant or the specific mag- 
netic rotation. The constant like the rotational constant a, de- 
pends on the wavelength. 


, 4 ^ ' 4 ^ 

Y Y 

(*) (*) 

Fig. 19.21 




Polarization of Light 


455 


The direction of rotation is determined by the direction of the 
magnetic field. The sign of rotation does not depend on the direction 
of the ray. Therefore, if we reflect the ray from a mirror and make it 
pass through the magnetized substance again in the opposite direc- 
tion, the rotation of the plane of polarization will double. 

The magnetic rotation of the polarization plane is due to the pre- 
cession of the electron orbits (see Sec. 7.7) produced under the ac- 
tion of the magnetic field. 

Optically active substances when acted upon by a magnetic field 
acquire an additional ability of rotating the plane of polarization 
that is added to their natural ability. 



CHAPTER 20 INTERACTIOM OF 

ELECTROMAGNETIC 

WAVES 

WITH A SUBSTANCE 

20.1. Dispersion of Light 

By the dispersion of light are meant phenomena due to the depen- 
dence of the refractive index of a substance on the length of the light 
wave. This dependence can be characterized by the function 

- f (K) (20.1) 

where X 0 is the length of a light wave in a vacuum. 

The derivative of n with respect to X 0 is called the dispersion of 
a substance. 

Function (20.1) for all transparent colourless substances in the 
visible part of the spectrum has the nature shown in Fig. 20.1. 

Diminishing of the wavelength is attended 
by an increase in the refractive index at a 
constantly growing rate. Hence, the dispersion 
of a substance dn/dX 0 is negative. Its absolute 
value increases when X 0 decreases. 

If a substance absorbs part of the rays, 
then the course of dispersion displays an ano- 
maly in the region of absorption and near it 
(see Fig. 20.6). On a certain section, the dis- 
persion of the substance dn!dX 0 will be posi- 
tive. Such a variation of n with X 0 is called 
anomalous dispersion. 

Media having the property of dispersion are known as dispersing 
ones. In these media, the speed of light waves depends on the wave- 
length X 0 or the frequency o. 



20.2. Group Velocity 

Strictly monochromatic light of the kind 

E = A cos (oat — kx + a) (20.2) 

is an infinite sequence in time and space of “crests” and “valleys” 
propagating along the x-axis with the phase velocity 


(20.3) 



Interaction of Electromagnetic Waves with a Substance 


457 


[see Eq. (14.9)]. We cannot use such a wave to transmit a signal be- 
cause each following crest differs in no way from the preceding one. 
To transmit a signal, we must put a “mark” on the wave, say, inter- 
rupt it for a certain time A t. In this case, however, the wave will 
no longer be described by Eq. (20.2). 

It is the simplest to transmit a signal with the aid of a light pulse 
(Fig. 20.2). According to the Fourier theorem, such a pulse can be 



represented as the superposition of waves of the kind given by Eq. 
(20.2) having frequencies confined within a certain interval A<o. 
A superposition of waves differing only slightly from one another in 
frequency is called a wave packet or a wave group. The analytical 
expression for a wave packet has the form 

<Do+Aa>/2 

E(x,t)— | ^4 (tt cos(a)f — tui+au) d<o (20.4) 

©0 — A ©/2 

(the subscript a> used with A, Ar, and a indicates that these quanti- 
ties differ for different frequencies). With a fixed value of f, a plot of 
function (20.4) has the form shown in Fig. 20.2. When t changes, 
the graph becomes displaced along the x-axis. Within the limits of 
a packet, plane waves amplify one another to a greater or smaller ex- 
tent. Outside these limits, they virtually completely annihilate one 
another. 

The relevant calculations show that the smaller the width of a 
packet Ax, the greater is the interval of frequencies Aco or according- 
ly the greater is the interval of wave numbers Ak needed to describe 
a packet with the aid of Eq. (20.4). The following relation holds: 

AkAx « 2 n (20.5) 

We must stress the fact that for the superposition of weaves described 
by Eq. (20.4) to be considered a wave packet, the condition 
A<o a> 0 must be obeyed. 


458 


Optics 


In a non-dispersing medium, all the plane waves forming a packet 
propagate with the same phase velocity v. It is evident that in this 
case the velocity of the packet coincides with v y and the shape of the 
packet does not change with time. It can be shown that a packet 
spreads in a dispersing medium with time — its width grows. If 
the dispersion is not great, spreading of the packet is not too fast. 
In this case, we can say that the packet travels with the velocity 
u, by which we mean the velocity of the centre of the packet, i.e. 
of the point with the maximum value of E. This velocity is called 



the group velocity. In a dispersing medium, the group veloc- 
ity u differs from the phase velocity v (here we mean the phase veloc- 
ity of the harmonic component with the maximum amplitude, in 
other words, the phase velocity for the dominating frequency). We 
shall show below that when dn/dX 0 <Z 0, the group velocity is smal- 
ler than the phase one (u<Zv); when dn/d\ 0 >0, the group veloc- 
ity is greater than the phase one ( u >i>). 

Figure 20.3 shows “photographs” of a wave packet for three con- 
secutive moments t ± , t 2y and t 3 . The figure is for the case when 
u < v. Inspection of the figure shows that motion of the packet is 
attended by motion of the crests and valleys “inside” it. New crests 
constantly appear at the left-hand boundary of the packet. After 
travelling along the packet, they vanish at its right-hand boundary. 
Hence, whereas the packet as a whole travels with the velocity u, 
the individual crests and valleys travel with the velocity v. 

When u the directions of motion of the packet and of the 

crests inside it are opposite. 

Let us explain what has been said above using the example of the 
superposition of two plane waves of the same amplitude and of different 
wavelengths X. Figure 20.4 gives an “instant photograph” of the 



Interaction of Electromagnetic Waves with a Substance 


459 


waves. One of them is shown by a solid line, and the other by a dash 
line. The intensity is the greatest at point A where the phases of the 
two waves coincide at the given moment. At points B and C, the 
two waves are in counterphase, owing to which the intensity of the 
resultant wave is zero. Assume that both waves are propagating from 
left to right, the velocity of the “solid” wave being lower than that 
of the “dash” one (here dv/dX >0 and, consequently, dn/dX < 0). 



Fig. 20.4 

Thus, the place at which the waves amplify each other will move to 
the left with time relative to the waves. As a result, the group veloc- 
ity will be lower than the phase value. If the velocity of the “solid” 
wave is greater than that of the “dash” one (i.e. dn!dX> 0), the place 
at which amplification of the waves occurs will move to the right 
so that the group velocity will be greater than the phase one. 

Let us write the equations of the waves, assuming for simplicity 
that the initial phases equal zero: 

E x = A cos (cof — kz) 

E % = A cos [(to + Aco) t — (k + Ak) x 1 

Here k = co/t^, and (k + A k) = (co + Acd)/z; 2 . Assume that Aco <C 
<C co, hence AAr <C k . Now, summating the oscillations and per- 
forming transformations according to the formula for the sum of co- 
sines, we get 

E = E t + E 2 = [2A cos (4p*— •— x)] cos ((of— kx) (20.6) 

(in the second multiplier, we have disregarded Aco in comparison 
with co and A k in comparison with &). 

The multiplier in brackets varies much more slowly with x and 
t than the second multiplier. We can therefore consider expression 
(20.6) as the equation of a plane wave whose amplitude varies ac- 
cording to the law* 

Amplitude = J 2 A cos ( t x J | 


• Compare with Eqs. (7.86) and (7.87) of Vol. I, p. 205. The dependence of 
function (20.6) on x at a fixed value of t is depicted by a curve similar to the one 
in Fig. 7.11a of Vol. I, p. 205. 



460 


Optics 


In the given case, there is a number of identical amplitude maxima 
determined by the condition 

t — y^x mai =±mji (m = 0, 1, 2, ...) (20.7) 


Each of these maxima can be considered as the centre of the relevant 
wave packet. 

Solving Eq. (20.7) relative to x max , we get 

Ag> . 4 

Xmax ^TT* + con9t 

It thus follows that the maxima travel with the velocity 



( 20 . 8 ) 


The expression obtained is the group velocity for a packet formed by 
two components. 

Let us find the velocity with which the centre of a wave packet 
described by expression (20.5) travels. Passing over from cosines to 
exponents, we get 

cd 0 +A(i>/2 

E(x,t) = ^ A a exp [i (©f — k a x)]d(a (20.9) 

fi>o — A©/2 

lila, = A a exp (ia tt ) is the complex amplitude]. 

Let us expand the function k a = k (©) into a series in the vicini- 
ty of © 0 : 

*» = *<>+ (l5-)o (a) ~ a)o) +'-' <20.10) 


Here k 0 — k (©) 0 , and (dk/d<a) 0 is the value of the derivative at 
point ©„. 

We shall introduce the variable £ = to — © 0 . Hence, © = © 0 + 
-j- £ and d© = d£. Performing such a substitution in Eq. (20.9) 
and introducing the value of k a from Eq. (20.10), we can write 


+ A < j >/2 

E (x, t) = exp [i (© 0 t — kox)] j /4 t exp 

-A*>/2 


( 20 . 11 ) 


We have arrived at an equation of a plane wave of frequency © 0 , 
wave number k 0 , and complex amplitude 

+A©/2 

A(*. 0- J -4,<*p{i[<-(£) 0 *]l}.i5 (20 '2) 

~A<i>/2 




Interaction of Electromagnetic Waves with a Substance 


461 


It can be seen from Eq. (20.12) that the equation 

*“(H)o a:==const ( 2013 ) 

relates the time t and the coordinate x of the plane in which the com- 
plex amplitude has a given fixed value, in particular including a 
value such that the magnitude of the complex amplitude, i.e. the con- 
ventional amplitude A ( x , f), reaches a maximum. 

Taking into account that l/(dk/d<n) 0 = ( d<a/dk) 0 , we can write 
Eq. (20.13) in the form 

Xmax “ (~5fc ) o * ' ” const ' [ const' = (d ° /d 3 * )o j (20. 14) 

It follows from Eq. (20.14) that the place where the amplitude of 
a wave packet is maximum travels with the velocity ( da>/dk ) 0 . 
We thus arrive at the following expression for the group velocity: 

»“4r (20.15) 


(the subscript 0 is no longer needed and has been omitted). We pre- 
viously obtained a similar expression for a packet of two waves [see 
Eq. (20.8)1. 

We remind our reader that we have disregarded the terms of higher 
orders of smallness in expansion (20.10). In this approximation, the 
shape of the wave packet does not change with time. If we take into 
account the following terms of the expansion, then we get an expres- 
sion for the amplitude from which it follows that the width of a packet 
grows with time — a wave packet broadens. 

We can give a different form to the expression for the group veloc- 
ity. Substituting vk for co [see Eq. (20.3)1, we can write Eq. (20.14) 
as follows: 


u 


d (uk) 
dk 


i i dv 


(20.16) 


We shall further write 

dv dv dX 

~dk ~dX ~dk 

We find from the relation X — 2n/k that dX/dk = — 2n/A;* = 
= — XI k. Accordingly, dv/dk == — (dv/dX)(X/k). Using this value in 
Eq. (20.16), we get 

U = v —X~ (20.17) 

A glance at this formula shows that the group velocity u can be 
either smaller or greater than the phase velocity v. depending on the 
sign of dvld \ . In the absence of dispersion, dv/dk = 0, and the group 
velocity coincides with the phase one. 


462 


Optics 


The maximum of the intensity falls to the centre of a wave packet. 
Therefore, when the concept of group velocity has a meaning, the 
velocity of energy transfer by a wave equals the group velocity. 

The concept of group velocity may be applied only provided that 
the absorption of the wave energy in the given medium is not great. 
With considerable attenuation of the waves, the concept of group 
velocity loses its meaning. This occurs in the region of anomalous 
dispersion. In this region, the absorption is very great, and the concept 
of group velocity cannot be applied. 


20.3. Elementary Theory of Dispersion 

The dispersion of light can be explained on the basis of the electro- 
magnetic theory and the electron theory of a substance. For this pur- 
pose, we must consider the process of interaction of light with a sub- 
stance. The motion of the electrons in an atom obeys the laws of 
quantum mechanics. In particular, the concept of the trajectory of 
an electron in an atom loses all meaning. As Lorentz showed, how- 
ever, it is sufficient to restrict ourselves to the hypothesis on the ex- 
istence of electrons bound quasi-elastically within atoms for a qua- 
litative understanding of many optical phenomena. When brought 
out of their equilibrium position, such electrons will begin to oscil- 
late, gradually losing the energy of oscillation on the emission of elec- 
tromagnetic waves. As a result, the oscillations will be damped. The 
attenuation can be taken into account by introducing the “force of 
friction of emission” proportional to the velocity. 

When an electromagnetic wave passes through a substance, every 
electron experiences the action of the Lorentz force 

F = —eE — e [vB] = —eE — e\i 0 [vHl (20.18) 

[see Eq. (6.35); the charge of an electron is — e]. According to Eq. 
(15.23), the ratio of the magnetic and electric field strengths in a 
wave is HIE = V e 0 /p 0 * Hence, from Eq. (20.18), we get the fol- 
lowing value for the ratio of the magnetic and electric forces exerted 
on an electron 

Even if the amplitude a of electron oscillations reached a value of 
the order of 1 S (10 -10 m), i.e. of the order of an atom’s dimensions, 
the amplitude of the velocity of an electron a© would be about 10~ 1# X 
X 3 X 10 15 = 3 x 10 5 m/s [according to Eq. (16.6), © = 2 Jtv 
equals about 3 X 10 15 rad/s]. Thus, the ratio v/c is clearly less 
than 10 -3 so that we may disregard the second addend in Eq. (20.18). 




Interaction of Electromagnetic Waves with a Substance 


463 


We can thus consider that when an electromagnetic wave passes 
through a substance, every electron experiences the force 

F = — eE 0 cos (c ot + a) 

(a is a quantity determined by the coordinates of a given electron, 
and E 0 is the amplitude of the electric field strength of the wave). 

To simplify our calculations, we shall first disregard the attenua- 
tion due to emission. We shall subsequently take the attenuation 
into account by introducing the relevant corrections into the formu- 
las obtained. The equation of motion of an electron in this case has 
the form 

• • ^ 

r ©Jr =5 E 0 cos (<of + a) 


Isee Eq. (7.13) of Vol. I, p. 189]; <o 0 is the natural frequency of oscil- 
lations of an electron). Let us add — i (e/m) E 0 sin (o£ + a) to the 
right-hand side of this equation and thus pass over to the complex 
functions E and r: 

-g)r + tojr = — E 0 exp (i <ot) (20. 19) 

Here E 0 = E 0 exp (ia) is the complex amplitude of the electric 
field of a wave. 

We shall seek a solution of the equation in the form r = r 0 exp (icoJ), 
where r 0 is the complex amplitude of oscillations of an elec- 
tron. Accordingly, (Pr/dt 2 = — co 2 r 0 exp (imt). Introducing these 
expressions into Eq. (20.19) and cancelling out the common factor 
exp (icof), we arrive at the expression 

— co a r 0 + © Jr 0 = — — E 0 


whence 


— (elm) Eq 
cdJ — CD* 


Multiplying the equation obtained by exp (ico*), we obtain 


C (f \ _ — (e/m) E (Q 

' ' cog — CD* 


Finally, taking the real parts of the complex functions r and E y 
we find r as a function of t: 


r(t) = 


— (e/m)S(t) 

<D$ — <D* 


( 20 . 20 ) 


To simplify the problem, we shall consider that the molecules are 
non-polar. In addition, since the masses of nuclei are great in compa- 



464 


Optics 


rison with the mass of an electron, we shall ignore the displacements 
of the nuclei from the equilibrium positions under the action of the 
wave field. In this approximation, the dipole electric moment of a 
molecule can be represented in the form 


P (0 = 21 ffiRo. i + 5 e h l r o, k -+■ T k (*)J = 

i h 

= ( S 9<Ro. i + 5 <Vo. it} + S e k*h (0 = 

i h h 

= Po+S ( t ) = 53 (t) 

It fc 

where and R 0> * = charges and position vectors of the equilibrium 
positions of the nuclei 

e k and r 0fk = charge and position vector of the equilibrium 
position of the A:-th electron 
r k (t) = displacement of the fc-th electron from its equi- 
librium position under the action of the wave 
field 

p 0 = dipole moment of a molecule in the absence of 
a field, which is assumed to equal zero. 

All the r k (t)'s are collinear with E(£). We therefore obtain the 
following expression for the projection of p(f) onto the direction 
of E(t): 

p(t) = H e k r h (t) = 53 ( — e) r k ( t ) 

h h 

(we have taken into account that e k for all electrons is identical and 
equals — e). Let us introduce into this equation the value of r (*) 
from Eq. (20.20), taking into consideration that the electrons in 
a molecule have different natural frequencies o) 0 ,fc- As a result, we 
get 

W'>-2-srir=3r B <‘> < 20 - 21 > 


Let us denote the number of molecules in unit volume by the sym- 
bol N. The product Np(t) gives the polarization P(t) of a substance. 
According to Eqs. (2.20) and (2.5), the permittivity is 


e=l+X=l 


PJt) 

eoE (t) 


1-i 


N P(t) 

eo E(t) 


Using in this expression the ratio p(t)/E(t) obtained from Eq. 
(20.21) and substituting n z for e {see Eq. (16.3)1, we arrive at the 
formula 


n 2 



«*/»n 


“i.»— 


( 20 . 22 ) 


Interaction of Electromagnetic Waves with a Substance 


465 


At frequencies <o appreciably differing from all the natural fre- 
quencies <D 0 the sum in Eq. (20.22) will be small in comparison 
with unity, so that n 2 « 1. Near each of the natural frequencies, 
function (20.22) is interrupted: when <o tends to cd 0 ,* from the left, 
it becomes equal to +oo, and when it tends to <a 0tfe from the right, 
the function becomes equal to — oo (see the dash curves in Fig. 
20.5). Such a behaviour of function (20.22) is due to the fact that we 
have disregarded the friction of emission [we remind our reader that 




when friction is disregarded, the amplitude of the forced oscilla- 
tions in resonance becomes equal to infinity; see Eq. (7.128) of 
Vol. I, p. 219]. When the friction of emission is taken into consid- 
eration, we get the dependence of n 2 on <o depicted in Fig. 20.5 by 
the solid curve. 

Passing over from n 2 to n and from co to X 0 , we get the curve shown 
in Fig. 20.6 (the. figure gives only a portion of the curve in the region 
of one of the resonance wavelengths). The dash curve in this figure 
shows how the coefficient of absorption of light by a substance changes 
(see the following section). Segment 3-4 is similar to the curve 
shown in Fig. 20.1. Segments 1-2 and 3-4 correspond to normal dis- 
persion (dn/dX 0 < 0). On segment 2-3, the dispersion is anomalous 
(dn/dX 0 >0). In region 1-2 , the refractive index is less than unity, 
hence, the phase velocity of the wave exceeds c. This circumstance 
does not contradict the theory of relativity, which is based on the 
statement that the velocity of transmitting a signal cannot exceed c. 
In the preceding section, we found that it is impossible to transmit 
a signal with the aid of an ideally monochromatic wave. Energy 
(i.e. a signal) is transmitted with the aid of a not completely mono- 
chromatic wave (wave packet), however, with a velocity equal to 
the group velocity determined by Eq. (20.17). In the region of nor- 
mal dispersion, du/dX >0 ( dn and du have different signs, while 
dn/dX < 0), so that although v Z>c, the group velocity is less than 
c. In the region of anomalous dispersion, the concept of group veloc- 
ity loses its meaning (the absorption is very great). Therefore, the 


466 


Optics 


value of u calculated by Eq. (20.17) will not characterize the rate of 
energy transmission. The relevant calculations give a value less 
than c for the velocity of energy transmission in this case too. 


20.4. Absorption of Light 

When a light wave passes through a substance, part of the wave 
energy is spent for producing oscillations of the electrons. This energy 
is partly returned to the radiation in the form of the secondary wave- 
lets set up by the electrons; it is partly transformed, however, into 
the energy of motion of the atoms, i.e. into the internal energy of 
the substance. This is the reason why the intensity of light transmitted 
through a substance diminishes — light is absorbed in the substance. 
The forced oscillations of the electrons and therefore the absorp- 
tion of light become especially intensive at the resonance frequency 
(see the dash absorption curve in Fig. 20.6). 

Experiments show that the intensity of light when it passes through 
a substance diminishes according to the exponential law 

/ = (20.23) 

Here J 0 = intensity of light at the entrance to the absorbing layer 
(on its boundary or at a certain place inside the sub- 
stance) 

l = thickness of the layer 

x = constant depending on the properties of the absorbing 
substance and called the absorption coefficient. 

Equation (20.23) is known as Bouguer’s law [in honour of the French 
scientist Pierre Bouguer (1698-1758)]. 

Differentiation of Eq. (20.23) yields 

dl = — xl 0 e~ yl dl= — xl dl (20.24) 

It follows from this expression that the decrement of the intensity 
along the path dl is proportional to the length of this path and 
to the value of the intensity itself. The absorption coefficient is the 
constant of proportionality. 

Inspection of Eq. (20.23) shows that when l = 1/x, the intensity 
I is i/e-th of J 0 . Thus, the absorption coefficient is a quantity inver- 
sely proportional to the thickness of the layer that reduces the inten- 
sity of light passing through it to 1/e-th of its initial value. 

The absorption coefficient depends on the wavelength X (or the 
frequency <d). The absorption coefficient of a substance whose atoms 
or molecules do not virtually act on one another (gases and metal 
vapours at a low pressure) is close to zero for most wavelengths. It 
displays sharp maxima (Fig. 20.7) only for very narrow spectral 
regions (having a width of several hundredths of an angstrom). These 




Interaction of EUctr^ma^netic Waves with a Substance 


467 


maxima correspond to the resonance frequencies of oscillations of the 
electrons inside the atoms. For polyatomic molecules, frequencies 
corresponding to the oscillations of the atoms inside the molecules 
are also detected. Since the masses of atoms are tens of thousands of 
times greater than the mass of an electron, the molecular frequencies 
are much smaller than the atomic ones— they are in the infrared re- 
gion of the spectrum. 

Gases at high pressures, and also liquids and solids produce broad 
absorption bands (Fig. 20.8). As the pressure of gases is increased, the 



Fig. 20.7 Fig. 20.8 

absorption maxima, which are initially very narrow (see Fig. 20.7), 
expand more and more, and at high pressures the absorption spec- 
trum of gases approaches those of liquids. This fact indicates that 
the expansion of the absorption bands is the result of the atoms in- 
teracting with one another. 

Metals are virtually opaque for light (x for them has a value of the 
order of 10 6 reciprocal metres; for comparison we shall point out 
that for glass x » 1 m -1 ). This is due to the presence of free electrons 
in metals. The action of the electric field of a light wave causes the 
free electrons to come into motion — fast-varying currents attended 
by the liberation of Lenz-Joule heat are produced in the metal. As 
a result, the energy of the light wave rapidly diminishes and trans- 
forms into the internal energy of the metal. 


20.5. Scattering of Light 

From the classical viewpoint, the process of scattering of light con- 
sists in that light passing through a substance causes the electrons 
in the atoms to oscillate. The oscillating electrons produce secondary 
wavelets that propagate in all directions. This phenomenon should 
seem to result in the scattering of light in all conditions. The secon- 
dary wavelets, however, are coherent, so that their mutual inter- 
ference must be taken into consideration. 

The relevant calculations show that in a homogeneous medium the 
secondary wavelets completely destroy one another in all directions 
except for that of propagation of the primary wave. Therefore, no 



468 


Optict 


redistribution of the light by directions y i.e. scattering of the light, 
occurs. 

The secondary wavelets do not destroy one another in side direc- 
tions only when light propagates in a non-homogeneous medium. 
The light waves become diffracted on the non-homogeneities of the 
medium and produce a diffraction pattern characterized by a quite 
uniform distribution of the intensity between all directions. Such 
diffraction on fine non-homogeneities is called the scattering of light. 

Media having a clearly expressed optical non-homogeneity are 
known as turbid media. They include (1) smoke, i.e. a suspension of 
very minute solid particles in a gas, (2) fogs and mists — suspensions 

of very minute liquid droplets in gases, 

(3) suspensions formed by solid par- 
ticles in the bulk of a liquid, 

(4) emulsions, i.e. suspensions of very 
minute droplets of One liquid in anoth- 
er one that does not dissolve the first 
liquid (an example of an emulsion is 
milk, which is a suspension of droplets 
of fat in water), and (5) solids such as 
mother-of-pearl, opals, and milk glass. 

Fig. 20.9 Light scattered on particles whose 

size is considerably smaller than the 
length of a light wave becomes partly polarized. The expla- 
nation is that the oscillations of the electrons produced by the scat- 
tered light beam occur in a plane at right angles to the beam (Fig. 
20.9). The oscillations of the vector E in a secondary wavelet occur 
in a plane passing through the direction of oscillations of the charges 
(see Fig. 15.6). Therefore, the light scattered by the particles in di- 
rections normal to the beam will be completely polarized. The scat- 
tered light is polarized only partly in directions that make an angle 
otfier than a right one with the beam. 

As a result of scattering of the light in side directions, the intensi- 
ty in the direction of its propagation diminishes more rapidly than 
when only absorption occurs. Consequently, for a turbid substance, 
Eq. (20.23) must contain the coefficient x' due to scattering in ad- 
dition to the absorption coefficient x: 

/ = J 0 e-(K+x')* (20.25) 

The constant x' is called the extinction coefficient. 

If the dimensions of the non-homogeneities are small in compari- 
son with the length of a light wave (not over ^ 0.1k), then the 
intensity of the scattered light I is proportional to the fourth power 
of the frequency or is inversely proportional to the fourth power of 
the wavelength: 

I OC CO 4 OC 


Scattered 



(20.26) 


Interaction of Electromagnetic Waves with a Substance 


469 


This relation is known as Rayleigh’s law after the British physicist 
John Rayleigh (1842-1919). It is easy to understand its origin if we 
take into account that the radiant power of an oscillating charge is 
proportional to the fourth power of the frequency and, consequently, 
is inversely proportional to the fourth power of the wavelength [see 
expression (15.46)]. 

If the dimensions of the non-homogeneities are comparable with 
the length of a wave, then the electrons at different spots on the non- 
homogeneities oscillate with an appreciable phase shift. This cir- 
cumstance makes the phenomenon more complicated and leads to 


1**0 



Fig. 20.10 

other regularities — the intensity of the scattered light becomes pro- 
portional to only the square of the frequency (inversely proportional 
to the square of the wavelength). 

It is simple to observe the manifestation of law (20.26) by passing 
a beam of white light through a vessel with a turbid liquid (Fig. 
20.10). Owing to scattering, the trace of the beam in the liquid is 
seen very well from a side. Since short light waves are scattered to 
a much greater extent than the long ones, the trace seems to be bluish. 
The beam passing through the liquid is enriched with long-wave 
radiation and forms a reddish-yellow spot on screen Sc instead of 
a white one. If we put polarizer P at the entrance of the beam to the 
vessel, we shall find that the intensity of the scattered light in differ- 
ent directions perpendicular to the initial beam is not the same. 
The directivity of dipole emission (see Fig. 15.7) results in the fact 
that in the directions coinciding with the plane of oscillations of 
the primary beam, the intensity of the scattered light virtually 
equals zero, while in the directions perpendicular to the plane of the 
oscillations, the intensity of the scattered light is maximum. By 
turning the polarizer around the direction of the primary beam, we 
shall observe alternate amplification and attenuation of the light 
scattered in the given direction. 

Even liquids and gases carefully purified of foreign admixtures 
and impurities scatter light to some extent. The Soviet physicist 
Leonid Mandelshtam (1879-1944) and the Polish physicist Marian 



470 


Optics 


Smoluchowski (1872-1917) established that the appearance of the 
optical non-homogeneities is due in this case to fluctuation of the 
density (i.e. deviations of the density from its mean value observed 
within the confines of small volumes). These fluctuations are pro- 
duced by chaotic motion of the molecules of the substance; therefore, 
the scattering of light due to them is called molecular. 

Molecular scattering explains the light blue colour of the sky. 
The places of compression and rarefaction of the air continuously 
appearing in the atmosphere owing to the random motion of its 
molecules scatter sunlight. According to law (20.26), the light blue 
and blue rays are scattered to a greater extent than the yellow and 
red ones, the result being the light blue colour of the sky. When the 
Sun is low above the horizon, the rays propagating directly from it 
pass through a scattering medium of great thickness, and as a result 
they are enriched with waves of greater lengths. This is why the 
sky at sunrise and sunset has red tints. 

There are especially favourable conditions for the appearance of 
considerable density fluctuations near the critical state of a substance 
(at the critical point dp/dV = 0; see Sec. 15.4 of Vol. I, p. 392). 
These fluctuations result in intensive scattering of light such that 
a glass ampule with the substance seems to be absolutely black when 
looked through. This phenomenon is known as critical opalescence. 


20.6. The Vavilov-Cerenkov Effect 

In 1934, the Soviet physicist Pavel Cerenkov (born 1904), working 
under the supervision of Sergei Vavilov (1891-1951), discovered a 
special kind of glow of liquids under the action of radium gamma- 
rays. Vavilov advanced the correct assumption that the fast electrons 
produced by the gamma-rays are the source of the radiation. This 
phenomenon was named the Vavilov-Cerenkov effect. Its complete 
theoretical explanation was given in 1937 by the Soviet physicists 
Igor Tamm (1895-1971) and Ilya Frank (born 1908)*. 

According to the electromagnetic theory, a charge moving uniform- 
ly emits no electromagnetic waves (see Sec. 15.6). As Tamm and 
Frank showed, however, this holds only if the velocity v of a charged 
particle does not exceed the phase velocity c/n of electromagnetic 
waves in the medium in which the particle is moving. A particle 
emits electromagnetic waves even when travelling uniformly pro- 
vided that v > c/n. The particle actually loses energy on radiation 
owing to which it travels with a negative acceleration. This accele- 
ration is not the cause (as when v < c/n), but a consequence of radia- 


* In 1958, Cerenkov, Tamm, and Frank were awarded a Nobel prize for 
their work. 



Interaction of Electromagnetic Waves with a Substance 


471 


tion. If the loss of energy at the expense of radiation were replen- 
ished in some way or other, a particle travelling uniformly with the 
velocity v >c/n would nevertheless be a source of radiation. 

The Vavilov-Cerenkov effect was observed experimentally for elec- 
trons, protons, and mesons travelling in liquid and solid media. 

Vavilov-Cerenkov radiation has a light blue colour because short 
waves predominate in it. The most characteristic feature of this ra- 
diation is the fact that it is emitted not in all directions, but only 



along the generatrices of a cone whose axis coincides with the direc- 
tion of velocity of the relevant particle (Fig. 20.11). The angle 
0 between the directions of propagation of the radiation and the ve- 
locity vector of a particle is determined by the equation 

COS0 — — (20.27) 

v nv ' 7 

The Vavilov-Cerenkov effect finds widespread application in ex- 
perimental equipment. In the so-called Cerenkov counters, a light 
pulse produced by a fast charged particle is transformed with the aid 
of a photomultiplier* into a current pulse. To make such a counter 
function, the energy of a particle must exceed the threshold value 
determined by the condition v = dn . Therefore, Cerenkov counters 
make it possible not only to register particles, but also to assess their 
energy. It is even possible to determine the angle 0 between the di- 
rection of a flash and the velocity of the particle. This allows us 
to use Eq. (20.27) to calculate the velocity (and, consequently, also 
the energy) of a particle. 


* By a photomultiplier is meant an electronic multiplier whose first ele- 
ctrode (a photocathode) is capable of emitting electrons under the action of light. 



CHAPTER 21 MOVING-MEDIA OPTICS 


21.1. The Speed of Light 

The speed of light in a vacuum is one of the fundamental physical 
quantities. The establishment of the finite nature of the speed of light 
had a tremendous significance of principle. The finite nature of the 
speed of transmitting signals and of transmitting interactions under- 
lies the theory of relativity. 

In view of the fact that the numerical value of the speed of light 
is very high, the experimental determination of this speed is a very 
complicated task. The speed of light was first 
determined on the basis of astronomical observa- 
tions. In 1676, the Danish astronomer Olaus 
Romer (1644-1710) determined the speed of light 
from observations of eclipses of Jupiter’s satel- 
lites. He obtained a value of 215 000 km/s. 

The Earth’s motion in orbit results in the 
visible position of stars on the celestial sphere 
changing. This phenomenon, called the aberra- 
tion of light, was used in 1727 by the British 
astronomer James Bradley (1693-1762) to deter- 
mine the speed of light. 

Assume that the direction to a star seen in 
a telescope is perpendicular to the plane of the 
Earth’s orbit. Hence, the angle between the 
direction toward the star and the vector of 
the Earth’s velocity v will be ji/ 2 during the 
entire year (Fig. 21.1). Let us point the axis 
of the telescope directly at the star. During 
Fig. 21.1 the time x needed for the light to cover the 

distance from the objective to the eyepiece, 
the telescope will move together with the Earth over the distance 
ut in a direction at right angles to the light ray. As a result, the im- 
age of the star will be displaced from the centre of the eyepiece. For 
the image to be exactly at the centre of the eyepiece, the axis of the 
telescope must be turned in the direction of the vector v throiigh the 
angle whose tangent is determined by the relation 

. v 

tan a= — 
c 



( 21 . 1 ) 



Moving-Media Optics 


473 


(see Fig. 21.1). In exactly the same way, raindrops falling vertically 
will fly through a long tube placed on a moving cart only if the axis 
of the tube is inclined in the direction of motion of the cart. 

Thus, the visible position of a star is displaced relative to the true 
one through the angle a. The Earth’s velocity vector constantly turns 
in the plane of the orbit. Therefore, the telescope axis also turns, de- 
scribing a cone about the true direction toward the star. Accordingly, 
the visible position of the star on the celestial sphere describes a cir- 
cle whose angular diameter is 2a. If the direction toward the star 


s 



makes an angle other than a right one with the plane of the Earth’s 
orbit, the visible position of the star describes an ellipse whose ma- 
jor axis has the angular dimension 2a. For a star in the plane of the 
orbit, the ellipse degenerates into a straight line. 

Bradley found from astronomical observations that 2a = 40.9". 
The corresponding value of c obtained by Eq. (21.1) is 303 000 km/s. 

In terrestrial conditions, the speed of light was first measured by 
the French scientist Armand Fizeau (1819-1896) in 1849. The layout 
of his experiment is shown in Fig. 21.2. Light from source S fell on 
a half-silvered mirror. The light reflected from the mirror got onto 
the edge of a rapidly rotating toothed disk. Every time a space be- 
tween the teeth was opposite the light beam, a light pulse was pro- 
duced that reached mirror M and was reflected back. If at the mo- 
ment when the light returned to the disk a space was opposite the 
beam, the reflected pulse passed partly through the half-silvered 
mirror and reached the observer’s eye. If a tooth of the disk was in 
the path of the reflected pulse, the observer saw no light. 

During the time t~2 He needed for the light to cover the distance 
to mirror M and back, the disk managed to turn through the angle 
Aq) = cox = 2/(d/c, where co is the angular velocity of the disk. 
Assume that the number of disk teeth is N . Therefore, the angle 
between the centres of adjacent teeth is a = 2 ji /N. The light did 
not return to the observer’s eye at such disk velocities at which the 
disk in the time t managed to turn through the angles a/2, 3a/2, 
..., (m — l/2)a, etc. Hence, the condition for the m-th blackout has 
the form 


A<P= ( 


m — 



a 


or 


2/co 


m 



!\ 2n 
2 ) AT 


c 



474 


Optics 


According to this formula, knowing Z, N , and the angular velocity 
(D m at which the m-th blackout is obtained, we can find c. In Fize- 
au’s experiment, Z was about 8.6 km. The value of 313 000 km/s 
was obtained for c . 

In 1928, Kerr cells (see Sec. 19.7) were used to measure the speed of 
light. They made it possible to interrupt a light beam with a much 
higher frequency (about 10 7 s" 1 ) than when a rotating toothed disk 
was used. This made measurements of c possible with l of the order of 
several metres. 

Albert Michelson performed several measurements of the speed of 
light using the method of a rotating prism. In Michelson’s experiment 
conducted in 1932, light propagated in a tube 1.6 km long from which 
the air was evacuated. 

At present, the speed of light in a vacuum is taken equal to 

c = 299 792.5 ± 0.1 km/s (21.2) 

We must note that in all the experiments in which light was in- 
terrupted, the group velocity of the light waves was determined, 
and not the phase velocity. In air, these two velocities virtually coin- 
cide. 


21.2. Fizeau’s Experiment 

Up to now, we assumed that the sources, receivers, and other bo- 
dies relative to which the propagation of light was considered are 
stationary. It is quite natural to be interested in how motion of a 
source of light waves affects the propagation of light. Here it becomes 
necessary to indicate relative to what the motion takes place. We 
established in Sec. 14.11 that the motion of a source or a receiver of 
sound waves relative to the medium in which these waves are pro- 
pagating affects the proceeding of acoustic phenomena (the Doppler 
effect), and, consequently, can be detected. 

The wave theory initially treated light as elastic waves propagat- 
ing in a hypothetic medium called universal ether. After Maxwell 
advanced his theory, elastic ether was replaced by an ether that was 
a carrier of electromagnetic waves and fields. By this ether was 
meant a special medium filling, like its elastic ether predecessor, the 
entire space of the universe and penetrating all bodies. Since ether 
was a certain medium, it would be possible to count on detecting the 
motion of bodies, for example light sources or receivers, with respect 
to this medium. In particular, the existence of an “ether wind” blow- 
ing around the Earth in its motion about the Sun ought to be ex- 
pected. 

Galileo’s principle of relativity was established in mechanics. 
According to it, all inertial reference frames are equivalent in a 



Moving-Media Optics 


475 


mechanical respect. The detection of ether would make it possible to 
separate (with the aid of optical phenomena) a special (related to 
ether) predominant, absolute reference frame. Therefore, motion 
of the other frames could be considered relative to this absolute 
frame. 

Thus, the establishment of how universal ether interacts with mov- 
ing bodies, was a matter of principle. Three possibilities could be 
assumed: (1) ether is absolutely not disturbed by moving bodies, 
(2) ether is partly carried along by moving bodies, acquiring a ve- 
locity of av y where v is the velocity of a body relative to the absolute 


% 

/ 

2 , 



H-** 

1 

■■Hi 

H 

' < $ Z J 

n naA 

2 ’ 1 

V 

u 


0 


Fig. 21.3 


reference frame, and a is a drag coefficient less than unity, and (3) 
ether is completely carried along by moving bodies, for example 
by the Earth, in the same way as a body in its motion carries along 
the layers of gas adjoining its surface. The last possibility, however, 
is disproved by the existence of the phenomenon of light aberration. 
We established in the preceding section that the change in the visible 
position of stars can be explained by the motion of the telescope re- 
lative to the reference frame (medium) in which the light wave is 
propagating. 

To find out whether ether is carried along by moving bodies, 
Fizeau conducted the following experiment in 1851. A parallel beam 
of light from source S was split by half-silvered plate Pinto two beams 
1 and 2 (Fig. 21.3). As a result of reflection from mirrors M x , M 2 
and Af 3 , the beams, after completing the same total path L, again 
reached plate P . Beam 1 partly passed through P, while beam 2 
was partly reflected. As a result, two coherent beams l f and 2 f were 
set up. They produced an interference pattern in the form of fringes 
in the focal plane of a telescope. Two tubes along which water could 
be passed with the velocity u in the directions indicated by the arrows 
were installed in the paths of beams 1 and 2. Ray 2 propagated in 
both tubes opposite to the flow of the water, and ray 1 with the flow. 



476 


Optics 


When the water was stationary, beams 1 and 2 covered the path 
L in the same time. If water in its motion even partly carries along 
ether, then when the flow of the water was switched on, ray 2 , which 
propagates opposite to the flow, would spend more time to cover the 
path L than ray 1 travelling in the direction of flow. As a result, a 
certain path difference will appear between the rays, and the inter- 
ference pattern will be displaced. 

The path difference we are interested in appears only in the path 
of the rays in the water. This path has the length 21. Let the velocity 
of light in the water relative to the ether be v. When ether is not car- 
ried along by the water, the speed of light relative to the arrangement 
will coincide with v. Let us assume that the water in its motion part- 
ly carries along the ether, imparting to it the velocity au relative to 
the arrangement ( u is the velocity of the water, and a is the drag 
coefficient). Hence, the velocity of light relative to the arrangement 
will be u + au for ray 1 and v — au for ray 2. Ray 1 covers the path 
21 during the time t x = 21/ (v + au) y and ray 2 during the time t 2 = 
= 21/ (u — au). It can be seen from Eq. (16.54) that the optical 
length of a path to cover which the time t is required equals ci . 
Hence, the path difference of rays 1 and 2 is A = c (t 2 — t x ). Divid- 
ing A by A, 0 , we get the number of fringes by which the interference 
pattern will be displaced when the flow of water is switched on: 

\7\J ___ C (*2 *i) c ( 2Z 21 \ __ 4 cIoax 

Ao Ao \ v — au i>+a u / Ao(i> 2 — oc*u 2 ) 

Fizeau discovered that the interference fringes are indeed dis- 
placed. The value of the drag coefficient corresponding to this dis- 
placement was 

a = l_JL (21.3) 


where n is the refractive index of water. Thus, Fizeau’ s experiment 
showed that ether (if it exists) is carried along by moving water only 
partly. 

It is easy to see that the result of Fizeau’s experiment is explained 
by the relativistic law of velocity addition. According to the first 
of equations (8.27) in Vol. I, p. 237, the velocities v x and of a body 
in frames K and K‘ are related by the expression 


<+ v o 


1 + 


(21.4) 


(y 0 is the velocity of the frame K ' relative to the frame K). 

Let us relate the reference frame K to Fizeau’s instrument, and 
the frame K' to the moving water. Now, the part of v 0 will be played 
by the velocity of the water u, that of v x by the velocity of the 
light relative to the water equal to c/n, and, finally, the part of v x 




Moving-Media Optics 


477 


will be played by the velocity of the light relative to the instrument 
v lnst . Introduction of these values into Eq. (21.4) yields 

c/n + tt c/n+w 

lnst i-\-u (cln)/c* 1 -f u/cn 

The velocity of the water u is much smaller than c. The expression 
obtained can therefore be simplified as follows: 

(■? + “)(*-■=■) -T+»(‘-w) < 21 ' 5 > 

(we have disregarded the term u 2 lcri). 

According to classical notions, the velocity of light relative to the 
instrument u ln8t equals the sum of the velocity of light relative to 
ether, i.e. c/n , and of the velocity of ether relative to the instrument, 
i.e. a u: 


V\nst = ^; + olu 

A comparison with Eq. (21.5) gives the value obtained by Fizeau 
for the drag coefficient a [see Eq. (21.3)1. 

It must be borne in mind that only the velocity of light in a va- 
cuum is the same in all reference frames. Its velocity in a substance 
differs in different reference frames. It has the value dn in the frame 
associated with the medium in which the light is propagating. 


21.3. Michelson’s Experiment 

In 1881, Michelson carried out his famous experiment by means of 
which he counted on detecting the motion of the Earth relative to 
etheF (the ether wind). In 1887, he repeated his experiment together 
with Morley on an improved instrument. The arrangement used by 
Michelson and Morley is shown in Fig. 21.4. A brick foundation sup- 
ported an annular iron trough with mercury. A wooden float having 
the shape of the bottom half of a longitudinally cut doughnut float- 
ed] on the mercury. The float carried a massive square stone slab. 
This design made it possible to smoothly turn the slab about the 
vertical axis of the arrangement. A Michelson interferometer (see 
Fig. 17.16) was installed on the slab. The interferometer was modi- 
fied so that both rays before returning to the half-silvered plate cover 
a distance coinciding with the diagonal of the slab several times. 
A diagram of the path of the rays is shown in Fig. 21.5. The symbols 
in this figure correspond to those used in Fig. 17.16. 

The experiment was based on the following reasoning. Let us as- 
sume that interferometer arm PM 2 (Fig. 21.6) coincides with the di- 








Moving-Media Optics 


479 


rection of motion of the Earth relative to ether. Consequently, the 
time needed for ray 1 to cover the path to mirror M x and back will 
differ from the time needed for ray 2 to cover path PM 2 P. As a re- 
sult, even when the lengths of both arms are equal, rays 1 and 2 
will acquire a certain path difference. If we turn the arrangement 
through 90 degrees, the arms will exchange places, and the path 
difference will change its sign. This should result in displacement of 
the interference pattern whose magnitude, as shown by calculations 
performed by Michelson, could be detected quite readily. 

To calculate the expected displacement of the interference pattern, 
let us find the time spent by rays 1 and 2 to cover the relevant paths. 






Assume that the Earth’s velocity relative to the ether is v. If the 
ether is not carried along by the Earth and the velocity of light re- 
lative to the ether is c (the refractive index of air is practically equal 
to unity), then the velocity of light relative to the instrument will 
be c — v for direction PM 2 and c + v for direction M 2 P • Hence, 
the time needed for ray 2 is determined by the expression 


l l 2lc 21 1 

c — v * c-\-v c a — v 2 c 1 — t^/c* 



( 21 . 6 ) 


(the Earth’s velocity along its orbit is 30 km/s, therefore v*/c 2 = 

= 10- 8 < 1 ). 

Before commencing to calculate the time f x , let us consider the fol- 
lowing example from mechanics. Suppose that a launch developing 
the velocity c relative to water has to cross a river with a current 
velocity of v in a direction strictly perpendicular to its banks (Fig. 
21.7). For the launch to travel in the required direction, its velocity 
c relative to the water must be directed as shown in the figure. There- 
fore, the velocity of the launch relative to the banks will be |c + 
+ v|= Y c % — v*. The velocity of ray 1 relative to the arrangement 
(as assumed by Michelson) will be the same. Consequently, the time 




480 


Optics 


taken by ray 1 is* 

t - 2Z __ 21 1 

1 Y C*—V 3 ~~ C Y 1— K»/C* 



(21.7) 


Substituting for and in the expression A = c (t t — t x ) their 
values from expressions (21.6) and (21.7), we get the path difference 
for rays 1 and 2: 

When the arrangement is turned through 90 degrees, the path differ- 
ence changes its sign. Consequently, the number of fringes by which 
the interference pattern will be displaced is 

A "=lr= 2 i!r-r (21 - 8 > 

The arm length l (taking into account multifold reflections) was 
11 m. The wavelength of the light used by Michelson and Morley 
was 0.59 pm. The use of these values in Eq. (21.8) gives 

A N = o/q^Vo-* - X 10- 8 = 0.37 « 0.4 fringe 

The arrangement made it possible to detect a displacement of the 
order of 0.01 fringe. But no displacement of the interference pattern 
was detected. The experiment was repeated during different times of 
the day to exclude the possibility of the horizon plane being perpen- 
dicular to the vector of the Earth’s orbital velocity at the moment of 
measurements. Subsequently, the experiment was repeated many 
times during different seasons of the year (during a year, the vector 
of the Earth’s orbital velocity turns in space through 360 degrees), 
and negative results were constantly obtained. The attempt to detect 
an ether wind was not successful. Universal ether remained elusive. 

Several attempts were made to explain the negative result of Mi- 
chelson ’s experiment without refuting the hypothesis of the existence 
of universal ether. But all these attempts were groundless. An ex- 
haustive non-contradictory explanation of all the experimental facts 
including the results of Michelson’s experiment was given by Albert 
Einstein in 1905. He arrived at the conclusion that universal ether, 
i.e. a special medium that could serve as an absolute reference frame, 
does not exist. Accordingly, Einstein extended the mechanical prin- 
ciple of relativity to all physical phenomena without any exception. 
He further postulated in accordance with experimental data that 
the speed of light in a vacuum is the same in all inertial reference 


* We have used the formulas V 1 — x » 1 — x/2 and 1/(1 — z) » 1 + 
* 11 ~ 



Moving-Media Optics 


481 


frames and does not depend on the motion of the light sources and 
receivers. 

The principle of relativity and the principle of the constancy of the 
speed of light form the foundation of the special theory of relativity 
developed by Einstein (see Chapter 8 of Vol. I). 


21.4. The Doppler Effect 


In acoustics, the change in frequency due to the Doppler effect 
is determined by the velocities of the source and the receiver rela- 
tive to the medium that is the carrier of the sound waves [see Eq. 


y 


n 


K 


K* 


Source Receiver x X* 


Fig. 21.8 


(14.78)]. The Doppler effect also exists for light waves. But there is 
no special medium that would serve as the carrier of electromagnetic 
waves. Therefore, the Doppler displacement of the frequency of 
light waves is determined only by the relative velocity of the source 
and the receiver. 

Let us associate the origin of coordinates of the frame K with a 
light source and the origin of coordinates of the frame K' with a re- 
ceiver (Fig. 21.8). We shall direct the axes x and x\ as usual, along 
the velocity vector v with which the frame K' (i.e. the receiver) is 
moving relative to the frame K (i.e. the source). The equation of a 
plane light wave emitted by the source in the direction of the re- 
ceiver will have the following form in the frame K : 

E (x, t) = Acos f<a (f — (21.9) 

Here <0 is the frequency of a wave registered in the reference frame as- 
sociated with the source, i.e. the frequency of oscillations of the 
source. We assume that the light wave is propagating in a vacuum; 
therefore the phase velocity is c. 

According to the principle of relativity, the laws of nature have 
the same form in all inertial reference frames. Hence, in the frame 
K\ the wave given by Eq. (21.9) will be described by the equation 

E{x', cos [©'(<' —j-)+ a'] (21.10) 



482 


Optics 


where ©' is the frequency registered in the reference frame K' , 
i.e. the frequency picked up by the receiver. We have provided all 
the quantities except c, which is the same in all reference frames, 
with primes. 

We can obtain an equation of a wave in the frame K' from an equa- 
tion in the frame /C by passing over from x and t to x' and t' with 
the aid of the Lorentz transformations. Introducing instead of x 
and t in Eq. (21.9) their values in accordance with Eqs. (8.17) of 
Vol. I, p. 229, we get 



x' -f- vt* "] 

t L yi_ v*/c* 

c y 1 — i */c* J 


a}